{"id":5379,"date":"2026-09-24T09:00:00","date_gmt":"2026-09-24T09:00:00","guid":{"rendered":"https:\/\/otus.ru\/journal\/?p=5379"},"modified":"2026-09-25T15:27:38","modified_gmt":"2026-09-25T15:27:38","slug":"teoriya-chisel-i-ih-razlozhenie","status":"publish","type":"post","link":"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/","title":{"rendered":"\u0420\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0447\u0438\u0441\u043b\u0430 \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u044b\u0435 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438: \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438, \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u0438 \u043a\u043e\u0434"},"content":{"rendered":"<p><strong>\u0421\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c<\/strong> &#8212; \u044d\u0442\u043e \u043a\u0430\u0436\u0434\u043e\u0435 \u0438\u0437 \u0447\u0438\u0441\u0435\u043b, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043f\u0435\u0440\u0435\u043c\u043d\u043e\u0436\u0430\u044e\u0442 \u0432 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0438: \u0432 \u0437\u0430\u043f\u0438\u0441\u0438 2 * 3 * 7 = 42 \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438 &#8212; 2, 3 \u0438 7. \u0420\u0430\u0437\u043b\u043e\u0436\u0438\u0442\u044c \u0447\u0438\u0441\u043b\u043e \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u044b\u0435 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438 &#8212; \u0437\u043d\u0430\u0447\u0438\u0442 \u0437\u0430\u043f\u0438\u0441\u0430\u0442\u044c \u0435\u0433\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435\u043c \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0435\u0439, \u043a\u0430\u0436\u0434\u044b\u0439 \u0438\u0437 \u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u043f\u0440\u043e\u0441\u0442\u043e\u0435 \u0447\u0438\u0441\u043b\u043e. \u041d\u0438\u0436\u0435 &#8212; \u0442\u0435\u043e\u0440\u0435\u043c\u0430, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u0433\u0430\u0440\u0430\u043d\u0442\u0438\u0440\u0443\u0435\u0442, \u0447\u0442\u043e \u0442\u0430\u043a\u043e\u0435 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0435\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u043e, \u0440\u0443\u0447\u043d\u044b\u0435 \u0441\u043f\u043e\u0441\u043e\u0431\u044b, \u043a\u043e\u0434 \u043d\u0430 Python \u0441 \u043e\u0446\u0435\u043d\u043a\u043e\u0439 \u0441\u043b\u043e\u0436\u043d\u043e\u0441\u0442\u0438 \u0438 \u0434\u0432\u0430 \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u044f: \u041d\u041e\u0414\/\u041d\u041e\u041a \u0438 \u043a\u0440\u0438\u043f\u0442\u043e\u0433\u0440\u0430\u0444\u0438\u044f RSA.<\/p>\n<div class=\"oj-toc\" style=\"background:#f6f6f6;border-radius:8px;padding:14px 20px;margin:20px 0\">\n<p><strong>\u0421\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435<\/strong><\/p>\n<ol>\n<li><a href=\"#s1\">\u041c\u0438\u043d\u0438-\u0441\u043b\u043e\u0432\u0430\u0440\u044c: \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c, \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c, \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u044c<\/a><\/li>\n<li><a href=\"#s2\">\u041e\u0441\u043d\u043e\u0432\u043d\u0430\u044f \u0442\u0435\u043e\u0440\u0435\u043c\u0430 \u0430\u0440\u0438\u0444\u043c\u0435\u0442\u0438\u043a\u0438<\/a><\/li>\n<li><a href=\"#s3\">\u041a\u0430\u043a \u0440\u0430\u0437\u043b\u043e\u0436\u0438\u0442\u044c \u0447\u0438\u0441\u043b\u043e \u0432\u0440\u0443\u0447\u043d\u0443\u044e<\/a><\/li>\n<li><a href=\"#s4\">\u0410\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u043f\u0435\u0440\u0435\u0431\u043e\u0440\u043e\u043c \u0434\u043e \u043a\u043e\u0440\u043d\u044f<\/a><\/li>\n<li><a href=\"#s5\">\u0421\u043b\u043e\u0436\u043d\u043e\u0441\u0442\u044c: \u043f\u043e\u0447\u0435\u043c\u0443 \u043f\u0435\u0440\u0435\u0431\u043e\u0440 \u043d\u0435 \u043c\u0430\u0441\u0448\u0442\u0430\u0431\u0438\u0440\u0443\u0435\u0442\u0441\u044f<\/a><\/li>\n<li><a href=\"#s6\">\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435 1: \u041d\u041e\u0414 \u0438 \u041d\u041e\u041a \u0447\u0435\u0440\u0435\u0437 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435<\/a><\/li>\n<li><a href=\"#s7\">\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435 2: \u043f\u043e\u0447\u0435\u043c\u0443 \u043d\u0430 \u0442\u0440\u0443\u0434\u043d\u043e\u0441\u0442\u0438 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u044f \u0434\u0435\u0440\u0436\u0438\u0442\u0441\u044f RSA<\/a><\/li>\n<li><a href=\"#s8\">\u0412\u044b\u0432\u043e\u0434\u044b<\/a><\/li>\n<li><a href=\"#s9\">\u0413\u0434\u0435 \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0435\u0442\u0441\u044f \/ \u0441\u0432\u044f\u0437\u044c \u0441 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u043e\u0439<\/a><\/li>\n<li><a href=\"#s10\">FAQ<\/a><\/li>\n<\/ol>\n<\/div>\n<h2 id=\"s1\">\u041c\u0438\u043d\u0438-\u0441\u043b\u043e\u0432\u0430\u0440\u044c: \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c, \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c, \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u044c<\/h2>\n<p>\u0422\u0435\u0440\u043c\u0438\u043d\u044b \u0431\u043b\u0438\u0437\u043a\u0438\u0435, \u043d\u043e \u043d\u0435 \u043e\u0434\u0438\u043d\u0430\u043a\u043e\u0432\u044b\u0435. \u0420\u0430\u0437\u0432\u0435\u0434\u0443 \u0438\u0445 \u043d\u0430 \u043e\u0434\u043d\u043e\u043c \u0447\u0438\u0441\u043b\u0435 42.<\/p>\n<table>\n<thead>\n<tr>\n<th>\u0422\u0435\u0440\u043c\u0438\u043d<\/th>\n<th>\u0427\u0442\u043e \u0437\u043d\u0430\u0447\u0438\u0442<\/th>\n<th>\u041f\u0440\u0438\u043c\u0435\u0440 \u0434\u043b\u044f 42<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u0421\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c (\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c)<\/td>\n<td>\u041b\u044e\u0431\u043e\u0435 \u0438\u0437 \u0447\u0438\u0441\u0435\u043b \u0432 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0438<\/td>\n<td>6 \u0438 7 \u0432 \u0437\u0430\u043f\u0438\u0441\u0438 6 * 7<\/td>\n<\/tr>\n<tr>\n<td>\u041f\u0440\u043e\u0441\u0442\u043e\u0439 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c<\/td>\n<td>\u0421\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043f\u0440\u043e\u0441\u0442\u044b\u043c \u0447\u0438\u0441\u043b\u043e\u043c<\/td>\n<td>2, 3, 7<\/td>\n<\/tr>\n<tr>\n<td>\u0414\u0435\u043b\u0438\u0442\u0435\u043b\u044c<\/td>\n<td>\u041d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e, \u043d\u0430 \u043a\u043e\u0442\u043e\u0440\u043e\u0435 42 \u0434\u0435\u043b\u0438\u0442\u0441\u044f \u0431\u0435\u0437 \u043e\u0441\u0442\u0430\u0442\u043a\u0430<\/td>\n<td>1, 2, 3, 6, 7, 14, 21, 42<\/td>\n<\/tr>\n<tr>\n<td>\u041a\u0430\u043d\u043e\u043d\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435<\/td>\n<td>\u041f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0441\u0442\u0435\u043f\u0435\u043d\u0435\u0439 \u043f\u0440\u043e\u0441\u0442\u044b\u0445 \u0432 \u043f\u043e\u0440\u044f\u0434\u043a\u0435 \u0432\u043e\u0437\u0440\u0430\u0441\u0442\u0430\u043d\u0438\u044f<\/td>\n<td>2 * 3 * 7<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u0421\u043b\u043e\u0432\u0430 \u00ab\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u00bb \u0438 \u00ab\u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u00bb \u0432 \u0441\u043e\u0432\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439 \u0448\u043a\u043e\u043b\u044c\u043d\u043e\u0439 \u0438 \u0431\u044b\u0442\u043e\u0432\u043e\u0439 \u0440\u0435\u0447\u0438 \u043f\u043e\u0447\u0442\u0438 \u0441\u0438\u043d\u043e\u043d\u0438\u043c\u044b. \u0412 \u043a\u043b\u0430\u0441\u0441\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0442\u0435\u0440\u043c\u0438\u043d\u043e\u043b\u043e\u0433\u0438\u0438 \u0443 \u0434\u0432\u0443\u0445 \u0447\u0438\u0441\u0435\u043b \u0432 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0438 \u0431\u044b\u043b\u0438 \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u044b\u0435 \u043d\u0430\u0437\u0432\u0430\u043d\u0438\u044f: \u00ab\u043c\u043d\u043e\u0436\u0438\u043c\u043e\u0435\u00bb \u0438 \u00ab\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u00bb, \u0430 \u00ab\u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438\u00bb &#8212; \u043e\u0431\u0449\u0435\u0435 \u0438\u043c\u044f \u0434\u043b\u044f \u043e\u0431\u043e\u0438\u0445. \u0412 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0445 \u0447\u0438\u0441\u043b\u0430\u0445 \u043b\u044e\u0431\u043e\u0439 \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0435\u0433\u043e \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u0435\u043c, \u043d\u043e \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u0435\u0439 \u043e\u0431\u044b\u0447\u043d\u043e \u0431\u043e\u043b\u044c\u0448\u0435, \u0447\u0435\u043c \u043f\u0440\u043e\u0441\u0442\u044b\u0445 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0435\u0439: 6 \u0438 14 \u0434\u0435\u043b\u044f\u0442 42, \u0430 \u0432 \u043a\u0430\u043d\u043e\u043d\u0438\u0447\u0435\u0441\u043a\u0443\u044e \u0437\u0430\u043f\u0438\u0441\u044c \u043d\u0435 \u0432\u0445\u043e\u0434\u044f\u0442.<\/p>\n<p><strong>\u041f\u0440\u043e\u0441\u0442\u043e\u0435 \u0447\u0438\u0441\u043b\u043e<\/strong> &#8212; \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e \u0431\u043e\u043b\u044c\u0448\u0435 1, \u0443 \u043a\u043e\u0442\u043e\u0440\u043e\u0433\u043e \u0440\u043e\u0432\u043d\u043e \u0434\u0432\u0430 \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u044f: 1 \u0438 \u043e\u043d\u043e \u0441\u0430\u043c\u043e. <strong>\u0421\u043e\u0441\u0442\u0430\u0432\u043d\u043e\u0435<\/strong> &#8212; \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e \u0431\u043e\u043b\u044c\u0448\u0435 1, \u0443 \u043a\u043e\u0442\u043e\u0440\u043e\u0433\u043e \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u0435\u0439 \u0431\u043e\u043b\u044c\u0448\u0435 \u0434\u0432\u0443\u0445. \u0427\u0438\u0441\u043b\u043e 1 \u043d\u0435 \u043f\u0440\u043e\u0441\u0442\u043e\u0435 \u0438 \u043d\u0435 \u0441\u043e\u0441\u0442\u0430\u0432\u043d\u043e\u0435.<\/p>\n<h2 id=\"s2\">\u041e\u0441\u043d\u043e\u0432\u043d\u0430\u044f \u0442\u0435\u043e\u0440\u0435\u043c\u0430 \u0430\u0440\u0438\u0444\u043c\u0435\u0442\u0438\u043a\u0438<\/h2>\n<p><strong>\u041b\u044e\u0431\u043e\u0435 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e n &gt; 1 \u0440\u0430\u0441\u043a\u043b\u0430\u0434\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u0432 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0441\u0442\u044b\u0445 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0435\u0439, \u0438 \u044d\u0442\u043e \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0435\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u043e \u0441 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u044c\u044e \u0434\u043e \u043f\u043e\u0440\u044f\u0434\u043a\u0430 \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0435\u0439.<\/strong><\/p>\n<p>\u041e\u0442\u0441\u044e\u0434\u0430 \u043a\u0430\u043d\u043e\u043d\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0437\u0430\u043f\u0438\u0441\u044c: n = p1^a1 * p2^a2 * &#8230; * pk^ak, \u0433\u0434\u0435 p1 &lt; p2 &lt; &#8230; &lt; pk &#8212; \u043f\u0440\u043e\u0441\u0442\u044b\u0435, \u0430 \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u0438 a1&#8230;ak &#8212; \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0435. \u041f\u0440\u0438\u043c\u0435\u0440\u044b:<\/p>\n<ul>\n<li>84 = 2^2 * 3 * 7;<\/li>\n<li>315 = 3^2 * 5 * 7;<\/li>\n<li>360 = 2^3 * 3^2 * 5;<\/li>\n<li>97 = 97 (\u043f\u0440\u043e\u0441\u0442\u043e\u0435 \u0447\u0438\u0441\u043b\u043e &#8212; \u044d\u0442\u043e \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0438\u0437 \u043e\u0434\u043d\u043e\u0433\u043e \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044f).<\/li>\n<\/ul>\n<p>\u041f\u043e\u0441\u043b\u0435\u0434\u043d\u0438\u0439 \u043f\u0440\u0438\u043c\u0435\u0440 \u0432\u0430\u0436\u0435\u043d: \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0435 \u043e\u0431\u044f\u0437\u0430\u043d\u043e \u0441\u043e\u0434\u0435\u0440\u0436\u0430\u0442\u044c \u0434\u0432\u0430 \u0438 \u0431\u043e\u043b\u044c\u0448\u0435 \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0435\u0439. \u0415\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0441\u0442\u044c, \u043a\u0441\u0442\u0430\u0442\u0438, &#8212; \u0433\u043b\u0430\u0432\u043d\u0430\u044f \u043f\u0440\u0438\u0447\u0438\u043d\u0430, \u043f\u043e \u043a\u043e\u0442\u043e\u0440\u043e\u0439 1 \u043d\u0435 \u0441\u0447\u0438\u0442\u0430\u044e\u0442 \u043f\u0440\u043e\u0441\u0442\u044b\u043c: \u0438\u043d\u0430\u0447\u0435 6 = 2 * 3 = 1 * 2 * 3 = 1 * 1 * 2 * 3, \u0438 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u043f\u0435\u0440\u0435\u0441\u0442\u0430\u043b\u043e \u0431\u044b \u0431\u044b\u0442\u044c \u0435\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u043c \u0434\u0430\u0436\u0435 \u0441 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u044c\u044e \u0434\u043e \u043f\u0435\u0440\u0435\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0438.<\/p>\n<h2 id=\"s3\">\u041a\u0430\u043a \u0440\u0430\u0437\u043b\u043e\u0436\u0438\u0442\u044c \u0447\u0438\u0441\u043b\u043e \u0432\u0440\u0443\u0447\u043d\u0443\u044e<\/h2>\n<p><strong>\u0421\u0442\u043e\u043b\u0431\u0438\u043a\u043e\u043c.<\/strong> \u0421\u043b\u0435\u0432\u0430 \u043f\u0438\u0448\u0443\u0442 \u0447\u0438\u0441\u043b\u043e, \u0441\u043f\u0440\u0430\u0432\u0430 &#8212; \u043d\u0430\u0438\u043c\u0435\u043d\u044c\u0448\u0438\u0439 \u043f\u0440\u043e\u0441\u0442\u043e\u0439 \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u044c, \u0434\u0435\u043b\u044f\u0442, \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u043f\u0435\u0440\u0435\u043d\u043e\u0441\u044f\u0442 \u0432\u043d\u0438\u0437 \u0438 \u043f\u043e\u0432\u0442\u043e\u0440\u044f\u044e\u0442, \u043f\u043e\u043a\u0430 \u0441\u043b\u0435\u0432\u0430 \u043d\u0435 \u043e\u0441\u0442\u0430\u043d\u0435\u0442\u0441\u044f 1. \u0414\u043b\u044f 84:<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\">84 | 2\n42 | 2\n21 | 3\n 7 | 7\n 1 |\n<\/code><\/pre>\n<p>\u041f\u0440\u0430\u0432\u0430\u044f \u043a\u043e\u043b\u043e\u043d\u043a\u0430 \u0434\u0430\u0435\u0442 \u043e\u0442\u0432\u0435\u0442: 84 = 2 * 2 * 3 * 7 = 2^2 * 3 * 7. \u041f\u043e\u0440\u044f\u0434\u043e\u043a \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0435\u0439 \u0432 \u043e\u0442\u0432\u0435\u0442\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f \u043d\u0435 \u0438\u043c\u0435\u0435\u0442, \u043d\u043e \u0432 \u043a\u0430\u043d\u043e\u043d\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0437\u0430\u043f\u0438\u0441\u0438 \u0438\u0445 \u0441\u0442\u0430\u0432\u044f\u0442 \u043f\u043e \u0432\u043e\u0437\u0440\u0430\u0441\u0442\u0430\u043d\u0438\u044e.<\/p>\n<p><strong>\u0414\u0435\u0440\u0435\u0432\u043e\u043c.<\/strong> \u0427\u0438\u0441\u043b\u043e \u0440\u0430\u0441\u043a\u043b\u0430\u0434\u044b\u0432\u0430\u044e\u0442 \u043d\u0430 \u043b\u044e\u0431\u0443\u044e \u043f\u0430\u0440\u0443 \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0435\u0439, \u0437\u0430\u0442\u0435\u043c \u043a\u0430\u0436\u0434\u0443\u044e \u0432\u0435\u0442\u043a\u0443 &#8212; \u0434\u0430\u043b\u044c\u0448\u0435, \u043f\u043e\u043a\u0430 \u0432 \u043b\u0438\u0441\u0442\u044c\u044f\u0445 \u043d\u0435 \u043e\u0441\u0442\u0430\u043d\u0443\u0442\u0441\u044f \u043f\u0440\u043e\u0441\u0442\u044b\u0435. \u0414\u043b\u044f 315: 315 = 9 * 35, \u0437\u0430\u0442\u0435\u043c 9 = 3 * 3 \u0438 35 = 5 * 7. \u041b\u0438\u0441\u0442\u044c\u044f 3, 3, 5, 7 \u0434\u0430\u044e\u0442 315 = 3^2 * 5 * 7. \u0415\u0441\u043b\u0438 \u043d\u0430\u0447\u0430\u0442\u044c \u0441 315 = 5 * 63, \u043b\u0438\u0441\u0442\u044c\u044f \u0431\u0443\u0434\u0443\u0442 \u0442\u0435 \u0436\u0435 &#8212; \u044d\u0442\u043e \u0438 \u0435\u0441\u0442\u044c \u0435\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0441\u0442\u044c \u0438\u0437 \u0442\u0435\u043e\u0440\u0435\u043c\u044b.<\/p>\n<p>\u041f\u0440\u043e\u0432\u0435\u0440\u044f\u0442\u044c \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u044c \u043d\u0430 2, 3 \u0438 5 \u0431\u044b\u0441\u0442\u0440\u043e \u043f\u043e\u043c\u043e\u0433\u0430\u044e\u0442 \u043f\u0440\u0438\u0437\u043d\u0430\u043a\u0438: \u043d\u0430 2 &#8212; \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u044f\u044f \u0446\u0438\u0444\u0440\u0430 \u0447\u0435\u0442\u043d\u0430\u044f, \u043d\u0430 3 &#8212; \u0441\u0443\u043c\u043c\u0430 \u0446\u0438\u0444\u0440 \u0434\u0435\u043b\u0438\u0442\u0441\u044f \u043d\u0430 3, \u043d\u0430 5 &#8212; \u0447\u0438\u0441\u043b\u043e \u043e\u043a\u0430\u043d\u0447\u0438\u0432\u0430\u0435\u0442\u0441\u044f \u043d\u0430 0 \u0438\u043b\u0438 5.<\/p>\n<h2 id=\"s4\">\u0410\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u043f\u0435\u0440\u0435\u0431\u043e\u0440\u043e\u043c \u0434\u043e \u043a\u043e\u0440\u043d\u044f<\/h2>\n<p>\u041a\u043b\u044e\u0447\u0435\u0432\u043e\u0435 \u043d\u0430\u0431\u043b\u044e\u0434\u0435\u043d\u0438\u0435: \u0435\u0441\u043b\u0438 n \u0441\u043e\u0441\u0442\u0430\u0432\u043d\u043e\u0435, \u0443 \u043d\u0435\u0433\u043e \u0435\u0441\u0442\u044c \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u044c \u043d\u0435 \u0431\u043e\u043b\u044c\u0448\u0435 \u221an. \u0415\u0441\u043b\u0438 n = a * b \u0438 \u043e\u0431\u0430 \u0431\u043e\u043b\u044c\u0448\u0435 \u221an, \u0442\u043e a * b &gt; n &#8212; \u043f\u0440\u043e\u0442\u0438\u0432\u043e\u0440\u0435\u0447\u0438\u0435. \u041f\u043e\u044d\u0442\u043e\u043c\u0443 \u043f\u0440\u043e\u0431\u043d\u044b\u0435 \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u0438 \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e \u043f\u0435\u0440\u0435\u0431\u0438\u0440\u0430\u0442\u044c, \u043f\u043e\u043a\u0430 d * d &lt;= n. \u0412\u0441\u0435, \u0447\u0442\u043e \u043e\u0441\u0442\u0430\u043b\u043e\u0441\u044c \u043f\u043e\u0441\u043b\u0435 \u044d\u0442\u043e\u0433\u043e \u0438 \u0431\u043e\u043b\u044c\u0448\u0435 1, &#8212; \u043f\u0440\u043e\u0441\u0442\u043e\u0435.<\/p>\n<p>\u041f\u0440\u043e\u0433\u043e\u043d \u0440\u0443\u043a\u0430\u043c\u0438 \u0434\u043b\u044f 84:<\/p>\n<ol>\n<li>d = 2: 84 -&gt; 42 -&gt; 21 (2 \u0432\u043e\u0448\u043b\u043e \u0434\u0432\u0430\u0436\u0434\u044b).<\/li>\n<li>d = 3: 3 * 3 = 9 &lt;= 21, 21 -&gt; 7.<\/li>\n<li>d = 5: 5 * 5 = 25 &gt; 7, \u0446\u0438\u043a\u043b \u0437\u0430\u043a\u0430\u043d\u0447\u0438\u0432\u0430\u0435\u0442\u0441\u044f.<\/li>\n<li>\u041e\u0441\u0442\u0430\u0442\u043e\u043a 7 &gt; 1 &#8212; \u044d\u0442\u043e \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0438\u0439 \u043f\u0440\u043e\u0441\u0442\u043e\u0439 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c.<\/li>\n<\/ol>\n<p>\u0422\u043e\u0442 \u0436\u0435 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u043d\u0430 Python (\u043f\u0440\u043e\u0432\u0435\u0440\u0435\u043d\u043e \u043d\u0430 Python 3.14):<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #C678DD\">def<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #61AFEF; font-weight: bold\">factorize<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">:<\/span> <span style=\"color: #E5C07B\">int<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #56B6C2\">-&gt;<\/span> <span style=\"color: #E5C07B\">dict<\/span><span style=\"color: #ABB2BF\">[<\/span><span style=\"color: #E5C07B\">int<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E5C07B\">int<\/span><span style=\"color: #ABB2BF\">]:<\/span>\n<span style=\"color: #ABB2BF\">    <\/span><span style=\"color: #98C379\">\"\"\"\u0420\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 n &gt;= 2 \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u044b\u0435 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438 \u043f\u0435\u0440\u0435\u0431\u043e\u0440\u043e\u043c \u0434\u043e \u043a\u043e\u0440\u043d\u044f.\"\"\"<\/span>\n    <span style=\"color: #C678DD\">if<\/span> <span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">&lt;<\/span> <span style=\"color: #D19A66\">2<\/span><span style=\"color: #ABB2BF\">:<\/span>\n        <span style=\"color: #C678DD\">raise<\/span> <span style=\"color: #E06C75\">ValueError<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #98C379\">\"\u043d\u0443\u0436\u043d\u043e \u0446\u0435\u043b\u043e\u0435 n &gt;= 2\"<\/span><span style=\"color: #ABB2BF\">)<\/span>\n    <span style=\"color: #E06C75\">factors<\/span><span style=\"color: #ABB2BF\">:<\/span> <span style=\"color: #E5C07B\">dict<\/span><span style=\"color: #ABB2BF\">[<\/span><span style=\"color: #E5C07B\">int<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E5C07B\">int<\/span><span style=\"color: #ABB2BF\">]<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #ABB2BF\">{}<\/span>\n    <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #D19A66\">2<\/span>\n    <span style=\"color: #C678DD\">while<\/span> <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">*<\/span> <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">&lt;=<\/span> <span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">:<\/span>\n        <span style=\"color: #C678DD\">while<\/span> <span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">%<\/span> <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">==<\/span> <span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">:<\/span>\n            <span style=\"color: #E06C75\">factors<\/span><span style=\"color: #ABB2BF\">[<\/span><span style=\"color: #E06C75\">d<\/span><span style=\"color: #ABB2BF\">]<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">factors<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">get<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">d<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #56B6C2\">+<\/span> <span style=\"color: #D19A66\">1<\/span>\n            <span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">\/\/=<\/span> <span style=\"color: #E06C75\">d<\/span>\n        <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">+=<\/span> <span style=\"color: #D19A66\">1<\/span> <span style=\"color: #C678DD\">if<\/span> <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">==<\/span> <span style=\"color: #D19A66\">2<\/span> <span style=\"color: #C678DD\">else<\/span> <span style=\"color: #D19A66\">2<\/span>   <span style=\"color: #7F848E\"># \u043f\u043e\u0441\u043b\u0435 2 \u043f\u0440\u043e\u0432\u0435\u0440\u044f\u0435\u043c \u0442\u043e\u043b\u044c\u043a\u043e \u043d\u0435\u0447\u0435\u0442\u043d\u044b\u0435<\/span>\n    <span style=\"color: #C678DD\">if<\/span> <span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">&gt;<\/span> <span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">:<\/span>                     <span style=\"color: #7F848E\"># \u043e\u0441\u0442\u0430\u0442\u043e\u043a \u0431\u043e\u043b\u044c\u0448\u0435 \u043a\u043e\u0440\u043d\u044f - \u043f\u0440\u043e\u0441\u0442\u043e\u0435 \u0447\u0438\u0441\u043b\u043e<\/span>\n        <span style=\"color: #E06C75\">factors<\/span><span style=\"color: #ABB2BF\">[<\/span><span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">]<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">factors<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">get<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #56B6C2\">+<\/span> <span style=\"color: #D19A66\">1<\/span>\n    <span style=\"color: #C678DD\">return<\/span> <span style=\"color: #E06C75\">factors<\/span>\n\n\n<span style=\"color: #C678DD\">def<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #61AFEF; font-weight: bold\">canonical<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">factors<\/span><span style=\"color: #ABB2BF\">:<\/span> <span style=\"color: #E5C07B\">dict<\/span><span style=\"color: #ABB2BF\">[<\/span><span style=\"color: #E5C07B\">int<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E5C07B\">int<\/span><span style=\"color: #ABB2BF\">])<\/span> <span style=\"color: #56B6C2\">-&gt;<\/span> <span style=\"color: #E5C07B\">str<\/span><span style=\"color: #ABB2BF\">:<\/span>\n    <span style=\"color: #E06C75\">parts<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #ABB2BF\">[]<\/span>\n    <span style=\"color: #C678DD\">for<\/span> <span style=\"color: #E06C75\">p<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">k<\/span> <span style=\"color: #56B6C2\">in<\/span> <span style=\"color: #E5C07B\">sorted<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">factors<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">items<\/span><span style=\"color: #ABB2BF\">()):<\/span>\n        <span style=\"color: #E06C75\">parts<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">append<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E5C07B\">str<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">p<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #C678DD\">if<\/span> <span style=\"color: #E06C75\">k<\/span> <span style=\"color: #56B6C2\">==<\/span> <span style=\"color: #D19A66\">1<\/span> <span style=\"color: #C678DD\">else<\/span> <span style=\"color: #98C379\">f\"{<\/span><span style=\"color: #E06C75\">p<\/span><span style=\"color: #98C379\">}^{<\/span><span style=\"color: #E06C75\">k<\/span><span style=\"color: #98C379\">}\"<\/span><span style=\"color: #ABB2BF\">)<\/span>\n    <span style=\"color: #C678DD\">return<\/span> <span style=\"color: #98C379\">\" * \"<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">join<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">parts<\/span><span style=\"color: #ABB2BF\">)<\/span>\n\n\n<span style=\"color: #C678DD\">for<\/span> <span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">in<\/span> <span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">84<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">315<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">360<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">97<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">1001<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">1024<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">600851475143<\/span><span style=\"color: #ABB2BF\">):<\/span>\n    <span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #98C379\">\"=\"<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">canonical<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">factorize<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">)))<\/span><\/code><\/pre>\n<p>\u0412\u044b\u0432\u043e\u0434:<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\">84 = 2^2 * 3 * 7\n315 = 3^2 * 5 * 7\n360 = 2^3 * 3^2 * 5\n97 = 97\n1001 = 7 * 11 * 13\n1024 = 2^10\n600851475143 = 71 * 839 * 1471 * 6857\n<\/code><\/pre>\n<p>\u0420\u0430\u0437\u0431\u043e\u0440. \u0412\u043d\u0443\u0442\u0440\u0435\u043d\u043d\u0438\u0439 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">while<\/code> \u0434\u0435\u043b\u0438\u0442 \u043d\u0430 d, \u043f\u043e\u043a\u0430 \u0434\u0435\u043b\u0438\u0442\u0441\u044f, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0432 \u0441\u043b\u043e\u0432\u0430\u0440\u044c \u043f\u043e\u043f\u0430\u0434\u0430\u044e\u0442 \u0442\u043e\u043b\u044c\u043a\u043e \u043f\u0440\u043e\u0441\u0442\u044b\u0435: \u043a \u043c\u043e\u043c\u0435\u043d\u0442\u0443 \u043f\u0440\u043e\u0432\u0435\u0440\u043a\u0438 d = 9 \u0432\u0441\u0435 \u0442\u0440\u043e\u0439\u043a\u0438 \u0443\u0436\u0435 \u0432\u044b\u043d\u0435\u0441\u0435\u043d\u044b, \u0438 9 \u043d\u0430 \u043e\u0441\u0442\u0430\u0442\u043e\u043a \u0443\u0436\u0435 \u043d\u0435 \u0434\u0435\u043b\u0438\u0442\u0441\u044f. \u0427\u0435\u0442\u043d\u044b\u0435 d \u043f\u043e\u0441\u043b\u0435 2 \u043a\u043e\u0434 \u0432\u043e\u043e\u0431\u0449\u0435 \u043d\u0435 \u043f\u0440\u043e\u0432\u0435\u0440\u044f\u0435\u0442, \u0430 \u0432 \u0432\u0430\u0440\u0438\u0430\u043d\u0442\u0435 \u0441 \u0448\u0430\u0433\u043e\u043c 1 (\u043d\u0438\u0436\u0435) d = 4 \u043d\u0435 \u0441\u0440\u0430\u0431\u043e\u0442\u0430\u0435\u0442 \u043f\u043e \u0442\u043e\u0439 \u0436\u0435 \u043f\u0440\u0438\u0447\u0438\u043d\u0435 &#8212; \u0434\u0432\u043e\u0439\u043a\u0438 \u0432\u044b\u043d\u0435\u0441\u0435\u043d\u044b \u0440\u0430\u043d\u044c\u0448\u0435. \u0423\u0441\u043b\u043e\u0432\u0438\u0435 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">d * d &lt;= n<\/code> \u043f\u0440\u043e\u0432\u0435\u0440\u044f\u0435\u0442\u0441\u044f \u043f\u043e \u0443\u0436\u0435 \u0443\u043c\u0435\u043d\u044c\u0448\u0435\u043d\u043d\u043e\u043c\u0443 n, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u043f\u043e\u0441\u043b\u0435 \u0432\u044b\u043d\u043e\u0441\u0430 \u043c\u0430\u043b\u044b\u0445 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0435\u0439 \u0446\u0438\u043a\u043b \u0437\u0430\u043a\u0430\u043d\u0447\u0438\u0432\u0430\u0435\u0442\u0441\u044f \u0440\u0430\u043d\u044c\u0448\u0435. \u0414\u0435\u043b\u0435\u043d\u0438\u0435 \u0446\u0435\u043b\u043e\u0447\u0438\u0441\u043b\u0435\u043d\u043d\u043e\u0435 (<code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">\/\/<\/code>), \u0431\u0435\u0437 \u043f\u0435\u0440\u0435\u0445\u043e\u0434\u0430 \u0432 float \u0438 \u043f\u043e\u0442\u0435\u0440\u0438 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u0438 \u043d\u0430 \u0431\u043e\u043b\u044c\u0448\u0438\u0445 \u0447\u0438\u0441\u043b\u0430\u0445.<\/p>\n<h3>\u0422\u0438\u043f\u0438\u0447\u043d\u0430\u044f \u043e\u0448\u0438\u0431\u043a\u0430: \u043f\u043e\u0442\u0435\u0440\u044f\u043d \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0438\u0439 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c<\/h3>\n<p>\u0415\u0441\u043b\u0438 \u0437\u0430\u0431\u044b\u0442\u044c \u0432\u0435\u0442\u043a\u0443 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">if n &gt; 1<\/code>, \u043f\u0440\u043e\u043f\u0430\u0434\u0430\u0435\u0442 \u043f\u0440\u043e\u0441\u0442\u043e\u0439 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c \u0431\u043e\u043b\u044c\u0448\u0435 \u043a\u043e\u0440\u043d\u044f \u0438\u0437 \u043e\u0441\u0442\u0430\u0442\u043a\u0430:<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #C678DD\">def<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #61AFEF; font-weight: bold\">factorize_bad<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">):<\/span>\n    <span style=\"color: #E06C75\">factors<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #ABB2BF\">{}<\/span>\n    <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #D19A66\">2<\/span>\n    <span style=\"color: #C678DD\">while<\/span> <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">*<\/span> <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">&lt;=<\/span> <span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">:<\/span>\n        <span style=\"color: #C678DD\">while<\/span> <span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">%<\/span> <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">==<\/span> <span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">:<\/span>\n            <span style=\"color: #E06C75\">factors<\/span><span style=\"color: #ABB2BF\">[<\/span><span style=\"color: #E06C75\">d<\/span><span style=\"color: #ABB2BF\">]<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">factors<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">get<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">d<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #56B6C2\">+<\/span> <span style=\"color: #D19A66\">1<\/span>\n            <span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">\/\/=<\/span> <span style=\"color: #E06C75\">d<\/span>\n        <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">+=<\/span> <span style=\"color: #D19A66\">1<\/span>\n    <span style=\"color: #C678DD\">return<\/span> <span style=\"color: #E06C75\">factors<\/span>          <span style=\"color: #7F848E\"># \u0437\u0430\u0431\u044b\u043b\u0438 \u043e\u0441\u0442\u0430\u0442\u043e\u043a n &gt; 1<\/span>\n\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">factorize_bad<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">14<\/span><span style=\"color: #ABB2BF\">),<\/span> <span style=\"color: #E06C75\">factorize_bad<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">97<\/span><span style=\"color: #ABB2BF\">),<\/span> <span style=\"color: #E06C75\">factorize_bad<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">84<\/span><span style=\"color: #ABB2BF\">))<\/span><\/code><\/pre>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\">{2: 1} {} {2: 2, 3: 1}\n<\/code><\/pre>\n<p>\u0414\u043b\u044f 14 \u043f\u043e\u0442\u0435\u0440\u044f\u043d\u0430 7, \u0434\u043b\u044f \u043f\u0440\u043e\u0441\u0442\u043e\u0433\u043e 97 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u0432\u043e\u043e\u0431\u0449\u0435 \u043f\u0443\u0441\u0442\u043e\u0439, \u0434\u043b\u044f 84 \u0441\u043d\u043e\u0432\u0430 \u043d\u0435\u0442 7. \u0418\u0441\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435 &#8212; \u0432\u0435\u0440\u043d\u0443\u0442\u044c \u043f\u0440\u043e\u0432\u0435\u0440\u043a\u0443 \u043e\u0441\u0442\u0430\u0442\u043a\u0430, \u043a\u0430\u043a \u0432 \u0440\u0430\u0431\u043e\u0447\u0435\u0439 \u0432\u0435\u0440\u0441\u0438\u0438 \u0432\u044b\u0448\u0435.<\/p>\n<h2 id=\"s5\">\u0421\u043b\u043e\u0436\u043d\u043e\u0441\u0442\u044c: \u043f\u043e\u0447\u0435\u043c\u0443 \u043f\u0435\u0440\u0435\u0431\u043e\u0440 \u043d\u0435 \u043c\u0430\u0441\u0448\u0442\u0430\u0431\u0438\u0440\u0443\u0435\u0442\u0441\u044f<\/h2>\n<p>\u0412 \u0445\u0443\u0434\u0448\u0435\u043c \u0441\u043b\u0443\u0447\u0430\u0435 (n \u043f\u0440\u043e\u0441\u0442\u043e\u0435 \u0438\u043b\u0438 \u0440\u0430\u0432\u043d\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044e \u0434\u0432\u0443\u0445 \u0431\u043b\u0438\u0437\u043a\u0438\u0445 \u043f\u0440\u043e\u0441\u0442\u044b\u0445) \u0446\u0438\u043a\u043b \u0434\u043e\u0445\u043e\u0434\u0438\u0442 \u0434\u043e \u221an. \u041f\u043e\u0441\u043b\u0435 \u0434\u0432\u043e\u0439\u043a\u0438 \u043f\u0440\u043e\u0432\u0435\u0440\u044f\u044e\u0442\u0441\u044f \u0442\u043e\u043b\u044c\u043a\u043e \u043d\u0435\u0447\u0435\u0442\u043d\u044b\u0435 d, \u044d\u0442\u043e \u043e\u043a\u043e\u043b\u043e \u221an \/ 2 \u0434\u0435\u043b\u0435\u043d\u0438\u0439. \u0417\u0430\u043c\u0435\u0440 \u043d\u0430 Python 3.14 \u0432 docker (\u0432\u0440\u0435\u043c\u044f \u0437\u0430\u0432\u0438\u0441\u0438\u0442 \u043e\u0442 \u043c\u0430\u0448\u0438\u043d\u044b):<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #C678DD\">import<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">time<\/span>\n\n<span style=\"color: #C678DD\">def<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #61AFEF; font-weight: bold\">trial_divisions<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">):<\/span>\n<span style=\"color: #ABB2BF\">    <\/span><span style=\"color: #98C379\">\"\"\"\u0421\u043a\u043e\u043b\u044c\u043a\u043e \u0434\u0435\u043b\u0435\u043d\u0438\u0439 \u0434\u0435\u043b\u0430\u0435\u0442 \u043f\u0435\u0440\u0435\u0431\u043e\u0440 (2, \u0437\u0430\u0442\u0435\u043c \u043d\u0435\u0447\u0435\u0442\u043d\u044b\u0435) \u0434\u043e \u043a\u043e\u0440\u043d\u044f.\"\"\"<\/span>\n    <span style=\"color: #E06C75\">count<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">2<\/span>\n    <span style=\"color: #C678DD\">while<\/span> <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">*<\/span> <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">&lt;=<\/span> <span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">:<\/span>\n        <span style=\"color: #E06C75\">count<\/span> <span style=\"color: #56B6C2\">+=<\/span> <span style=\"color: #D19A66\">1<\/span>\n        <span style=\"color: #C678DD\">while<\/span> <span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">%<\/span> <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">==<\/span> <span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">:<\/span>\n            <span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">\/\/=<\/span> <span style=\"color: #E06C75\">d<\/span>\n            <span style=\"color: #E06C75\">count<\/span> <span style=\"color: #56B6C2\">+=<\/span> <span style=\"color: #D19A66\">1<\/span>\n        <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">+=<\/span> <span style=\"color: #D19A66\">1<\/span> <span style=\"color: #C678DD\">if<\/span> <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">==<\/span> <span style=\"color: #D19A66\">2<\/span> <span style=\"color: #C678DD\">else<\/span> <span style=\"color: #D19A66\">2<\/span>\n    <span style=\"color: #C678DD\">return<\/span> <span style=\"color: #E06C75\">count<\/span>\n\n<span style=\"color: #C678DD\">for<\/span> <span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">in<\/span> <span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">10<\/span><span style=\"color: #56B6C2\">**<\/span><span style=\"color: #D19A66\">6<\/span> <span style=\"color: #56B6C2\">+<\/span> <span style=\"color: #D19A66\">3<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">10<\/span><span style=\"color: #56B6C2\">**<\/span><span style=\"color: #D19A66\">9<\/span> <span style=\"color: #56B6C2\">+<\/span> <span style=\"color: #D19A66\">7<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">10<\/span><span style=\"color: #56B6C2\">**<\/span><span style=\"color: #D19A66\">12<\/span> <span style=\"color: #56B6C2\">+<\/span> <span style=\"color: #D19A66\">39<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">10<\/span><span style=\"color: #56B6C2\">**<\/span><span style=\"color: #D19A66\">6<\/span> <span style=\"color: #56B6C2\">+<\/span> <span style=\"color: #D19A66\">3<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #56B6C2\">*<\/span> <span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">10<\/span><span style=\"color: #56B6C2\">**<\/span><span style=\"color: #D19A66\">6<\/span> <span style=\"color: #56B6C2\">+<\/span> <span style=\"color: #D19A66\">33<\/span><span style=\"color: #ABB2BF\">),<\/span> <span style=\"color: #D19A66\">2<\/span><span style=\"color: #56B6C2\">**<\/span><span style=\"color: #D19A66\">40<\/span><span style=\"color: #ABB2BF\">):<\/span>\n    <span style=\"color: #E06C75\">t<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">time<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">perf_counter<\/span><span style=\"color: #ABB2BF\">()<\/span>\n    <span style=\"color: #E06C75\">c<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">trial_divisions<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">)<\/span>\n    <span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #98C379\">f\"{<\/span><span style=\"color: #E06C75\">n<\/span><span style=\"color: #98C379\">:&gt;16}  \u0434\u0435\u043b\u0435\u043d\u0438\u0439: {<\/span><span style=\"color: #E06C75\">c<\/span><span style=\"color: #98C379\">:&gt;8}  \u0432\u0440\u0435\u043c\u044f: {<\/span><span style=\"color: #E06C75\">time<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">perf_counter<\/span><span style=\"color: #ABB2BF\">() <\/span><span style=\"color: #56B6C2\">-<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">t<\/span><span style=\"color: #98C379\">:.3f} c\"<\/span><span style=\"color: #ABB2BF\">)<\/span><\/code><\/pre>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\">         1000003  \u0434\u0435\u043b\u0435\u043d\u0438\u0439:      500  \u0432\u0440\u0435\u043c\u044f: 0.000 c\n      1000000007  \u0434\u0435\u043b\u0435\u043d\u0438\u0439:    15811  \u0432\u0440\u0435\u043c\u044f: 0.001 c\n   1000000000039  \u0434\u0435\u043b\u0435\u043d\u0438\u0439:   500000  \u0432\u0440\u0435\u043c\u044f: 0.084 c\n   1000036000099  \u0434\u0435\u043b\u0435\u043d\u0438\u0439:   500003  \u0432\u0440\u0435\u043c\u044f: 0.075 c\n   1099511627776  \u0434\u0435\u043b\u0435\u043d\u0438\u0439:       41  \u0432\u0440\u0435\u043c\u044f: 0.000 c\n<\/code><\/pre>\n<p>\u041f\u0435\u0440\u0432\u044b\u0435 \u0447\u0435\u0442\u044b\u0440\u0435 \u0447\u0438\u0441\u043b\u0430 &#8212; \u043f\u0440\u043e\u0441\u0442\u044b\u0435 \u0438\u043b\u0438 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0434\u0432\u0443\u0445 \u043f\u0440\u043e\u0441\u0442\u044b\u0445 \u043e\u043a\u043e\u043b\u043e \u043c\u0438\u043b\u043b\u0438\u043e\u043d\u0430, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u043f\u0435\u0440\u0435\u0431\u043e\u0440 \u0438\u0434\u0435\u0442 \u0434\u043e \u043a\u043e\u0440\u043d\u044f. \u0423 2^40 \u0432\u0441\u0435 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438 \u043c\u0430\u043b\u0435\u043d\u044c\u043a\u0438\u0435: 41 \u0434\u0435\u043b\u0435\u043d\u0438\u0435. \u041e\u0442\u0441\u044e\u0434\u0430 \u0433\u0440\u0430\u043d\u0438\u0446\u0430 \u043e\u0446\u0435\u043d\u043a\u0438: O(\u221an) &#8212; \u044d\u0442\u043e \u0445\u0443\u0434\u0448\u0438\u0439 \u0441\u043b\u0443\u0447\u0430\u0439, \u0430 \u043d\u0430 \u0447\u0438\u0441\u043b\u0430\u0445 \u0441 \u043c\u0430\u043b\u044b\u043c\u0438 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044f\u043c\u0438 \u043f\u0435\u0440\u0435\u0431\u043e\u0440 \u0437\u0430\u043a\u0430\u043d\u0447\u0438\u0432\u0430\u0435\u0442\u0441\u044f \u0431\u044b\u0441\u0442\u0440\u043e.<\/p>\n<p>\u0412\u0430\u0436\u043d\u0430\u044f \u0434\u0435\u0442\u0430\u043b\u044c: O(\u221an) \u0432\u044b\u0433\u043b\u044f\u0434\u0438\u0442 \u0441\u043a\u0440\u043e\u043c\u043d\u043e, \u043d\u043e \u0440\u0430\u0437\u043c\u0435\u0440 \u0432\u0445\u043e\u0434\u0430 &#8212; \u044d\u0442\u043e \u043a\u043e\u043b\u0438\u0447\u0435\u0441\u0442\u0432\u043e \u0446\u0438\u0444\u0440, \u0430 \u043d\u0435 \u0441\u0430\u043c\u043e n. \u0414\u043b\u044f \u0447\u0438\u0441\u043b\u0430 \u0438\u0437 b \u0431\u0438\u0442 \u221an = 2^(b\/2), \u0442\u043e \u0435\u0441\u0442\u044c \u0432\u0440\u0435\u043c\u044f \u0440\u0430\u0441\u0442\u0435\u0442 \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u043e \u043e\u0442 \u0434\u043b\u0438\u043d\u044b \u0437\u0430\u043f\u0438\u0441\u0438. \u041a\u0430\u0436\u0434\u044b\u0435 2 \u043b\u0438\u0448\u043d\u0438\u0445 \u0434\u0435\u0441\u044f\u0442\u0438\u0447\u043d\u044b\u0445 \u0437\u043d\u0430\u043a\u0430 \u0432 n \u0443\u0432\u0435\u043b\u0438\u0447\u0438\u0432\u0430\u044e\u0442 \u0445\u0443\u0434\u0448\u0438\u0439 \u0441\u043b\u0443\u0447\u0430\u0439 \u043f\u0440\u0438\u043c\u0435\u0440\u043d\u043e \u0432 10 \u0440\u0430\u0437.<\/p>\n<table>\n<thead>\n<tr>\n<th>\u0427\u0438\u0441\u043b\u043e n<\/th>\n<th>\u0426\u0438\u0444\u0440<\/th>\n<th>\u0414\u0435\u043b\u0435\u043d\u0438\u0439 \u0432 \u0445\u0443\u0434\u0448\u0435\u043c \u0441\u043b\u0443\u0447\u0430\u0435 (\u043e\u043a\u043e\u043b\u043e \u221an \/ 2)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>10^12<\/td>\n<td>13<\/td>\n<td>5 * 10^5<\/td>\n<\/tr>\n<tr>\n<td>10^18<\/td>\n<td>19<\/td>\n<td>5 * 10^8<\/td>\n<\/tr>\n<tr>\n<td>2^2048 (\u043c\u043e\u0434\u0443\u043b\u044c RSA)<\/td>\n<td>617<\/td>\n<td>\u043e\u043a\u043e\u043b\u043e 10^308<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u0414\u043b\u044f \u0447\u0438\u0441\u0435\u043b \u0434\u043e 10^18 \u043d\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u0435 \u0431\u0435\u0440\u0443\u0442 \u0431\u043e\u043b\u0435\u0435 \u0431\u044b\u0441\u0442\u0440\u044b\u0435 \u043c\u0435\u0442\u043e\u0434\u044b (\u0440\u043e-\u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u041f\u043e\u043b\u043b\u0430\u0440\u0434\u0430 \u043f\u043b\u044e\u0441 \u0442\u0435\u0441\u0442 \u043f\u0440\u043e\u0441\u0442\u043e\u0442\u044b \u041c\u0438\u043b\u043b\u0435\u0440\u0430 &#8212; \u0420\u0430\u0431\u0438\u043d\u0430), \u0432 Python \u043e\u043d\u0438 \u0435\u0441\u0442\u044c, \u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u0432 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">sympy.factorint<\/code>. \u0414\u043b\u044f \u0441\u043e\u0442\u0435\u043d \u0446\u0438\u0444\u0440 \u0438\u0437\u0432\u0435\u0441\u0442\u043d\u044b \u0442\u043e\u043b\u044c\u043a\u043e \u0441\u0443\u0431\u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u044b\u0435 \u043a\u043b\u0430\u0441\u0441\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u044b, \u0438 \u043d\u0430 \u043a\u043b\u044e\u0447\u0430\u0445 RSA \u0441\u043e\u0432\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439 \u0434\u043b\u0438\u043d\u044b \u0438\u0445 \u043d\u0435 \u0445\u0432\u0430\u0442\u0430\u0435\u0442.<\/p>\n<h2 id=\"s6\">\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435 1: \u041d\u041e\u0414 \u0438 \u041d\u041e\u041a \u0447\u0435\u0440\u0435\u0437 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435<\/h2>\n<p>\u0415\u0441\u043b\u0438 \u043e\u0431\u0430 \u0447\u0438\u0441\u043b\u0430 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u044b \u043a\u0430\u043d\u043e\u043d\u0438\u0447\u0435\u0441\u043a\u0438, <strong>\u041d\u041e\u0414<\/strong> &#8212; \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043e\u0431\u0449\u0438\u0445 \u043f\u0440\u043e\u0441\u0442\u044b\u0445 \u0432 \u043c\u0438\u043d\u0438\u043c\u0430\u043b\u044c\u043d\u043e\u0439 \u0441\u0442\u0435\u043f\u0435\u043d\u0438, <strong>\u041d\u041e\u041a<\/strong> &#8212; \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0432\u0441\u0435\u0445 \u043f\u0440\u043e\u0441\u0442\u044b\u0445 \u0432 \u043c\u0430\u043a\u0441\u0438\u043c\u0430\u043b\u044c\u043d\u043e\u0439 \u0441\u0442\u0435\u043f\u0435\u043d\u0438. \u041f\u0440\u0438\u043c\u0435\u0440: 196 = 2^2 * 7^2, 260 = 2^2 * 5 * 13. \u041e\u0431\u0449\u0435\u0435 \u043f\u0440\u043e\u0441\u0442\u043e\u0435 \u0442\u043e\u043b\u044c\u043a\u043e 2, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u041d\u041e\u0414 = 2^2 = 4, \u0430 \u041d\u041e\u041a = 2^2 * 5 * 7^2 * 13 = 12740.<\/p>\n<p>\u0412 Python <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">Counter<\/code> \u0434\u0435\u043b\u0430\u0435\u0442 \u044d\u0442\u043e \u043e\u043f\u0435\u0440\u0430\u0442\u043e\u0440\u0430\u043c\u0438: <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">&amp;<\/code> \u0431\u0435\u0440\u0435\u0442 \u043c\u0438\u043d\u0438\u043c\u0443\u043c \u0441\u0442\u0435\u043f\u0435\u043d\u0435\u0439, <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">|<\/code> &#8212; \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c:<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #C678DD\">from<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">collections<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #C678DD\">import<\/span> <span style=\"color: #E06C75\">Counter<\/span>\n<span style=\"color: #C678DD\">from<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">math<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #C678DD\">import<\/span> <span style=\"color: #E06C75\">gcd<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">lcm<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">prod<\/span>\n\n<span style=\"color: #C678DD\">def<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #61AFEF; font-weight: bold\">factorize<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">):<\/span>\n    <span style=\"color: #E06C75\">f<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">Counter<\/span><span style=\"color: #ABB2BF\">(),<\/span> <span style=\"color: #D19A66\">2<\/span>\n    <span style=\"color: #C678DD\">while<\/span> <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">*<\/span> <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">&lt;=<\/span> <span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">:<\/span>\n        <span style=\"color: #C678DD\">while<\/span> <span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">%<\/span> <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">==<\/span> <span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">:<\/span>\n            <span style=\"color: #E06C75\">f<\/span><span style=\"color: #ABB2BF\">[<\/span><span style=\"color: #E06C75\">d<\/span><span style=\"color: #ABB2BF\">]<\/span> <span style=\"color: #56B6C2\">+=<\/span> <span style=\"color: #D19A66\">1<\/span>\n            <span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">\/\/=<\/span> <span style=\"color: #E06C75\">d<\/span>\n        <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">+=<\/span> <span style=\"color: #D19A66\">1<\/span> <span style=\"color: #C678DD\">if<\/span> <span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">==<\/span> <span style=\"color: #D19A66\">2<\/span> <span style=\"color: #C678DD\">else<\/span> <span style=\"color: #D19A66\">2<\/span>\n    <span style=\"color: #C678DD\">if<\/span> <span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">&gt;<\/span> <span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">:<\/span>\n        <span style=\"color: #E06C75\">f<\/span><span style=\"color: #ABB2BF\">[<\/span><span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">]<\/span> <span style=\"color: #56B6C2\">+=<\/span> <span style=\"color: #D19A66\">1<\/span>\n    <span style=\"color: #C678DD\">return<\/span> <span style=\"color: #E06C75\">f<\/span>\n\n<span style=\"color: #C678DD\">def<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #61AFEF; font-weight: bold\">from_factors<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">f<\/span><span style=\"color: #ABB2BF\">):<\/span>\n    <span style=\"color: #C678DD\">return<\/span> <span style=\"color: #E06C75\">prod<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">p<\/span> <span style=\"color: #56B6C2\">**<\/span> <span style=\"color: #E06C75\">k<\/span> <span style=\"color: #C678DD\">for<\/span> <span style=\"color: #E06C75\">p<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">k<\/span> <span style=\"color: #56B6C2\">in<\/span> <span style=\"color: #E06C75\">f<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">items<\/span><span style=\"color: #ABB2BF\">())<\/span>\n\n<span style=\"color: #E06C75\">a<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">b<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">factorize<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">196<\/span><span style=\"color: #ABB2BF\">),<\/span> <span style=\"color: #E06C75\">factorize<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">260<\/span><span style=\"color: #ABB2BF\">)<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E5C07B\">dict<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">a<\/span><span style=\"color: #ABB2BF\">),<\/span> <span style=\"color: #E5C07B\">dict<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">b<\/span><span style=\"color: #ABB2BF\">))<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #98C379\">\"\u041d\u041e\u0414:\"<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">from_factors<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">a<\/span> <span style=\"color: #56B6C2\">&amp;<\/span> <span style=\"color: #E06C75\">b<\/span><span style=\"color: #ABB2BF\">),<\/span> <span style=\"color: #E06C75\">gcd<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">196<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">260<\/span><span style=\"color: #ABB2BF\">))<\/span>   <span style=\"color: #7F848E\"># &amp; - \u043c\u0438\u043d\u0438\u043c\u0443\u043c \u0441\u0442\u0435\u043f\u0435\u043d\u0435\u0439<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #98C379\">\"\u041d\u041e\u041a:\"<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">from_factors<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">a<\/span> <span style=\"color: #56B6C2\">|<\/span> <span style=\"color: #E06C75\">b<\/span><span style=\"color: #ABB2BF\">),<\/span> <span style=\"color: #E06C75\">lcm<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">196<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">260<\/span><span style=\"color: #ABB2BF\">))<\/span>   <span style=\"color: #7F848E\"># | - \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c \u0441\u0442\u0435\u043f\u0435\u043d\u0435\u0439<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #98C379\">\"\u041d\u041e\u0414\/\u041d\u041e\u041a 15 \u0438 40:\"<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">from_factors<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">factorize<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">15<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #56B6C2\">&amp;<\/span> <span style=\"color: #E06C75\">factorize<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">40<\/span><span style=\"color: #ABB2BF\">)),<\/span>\n      <span style=\"color: #E06C75\">from_factors<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">factorize<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">15<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #56B6C2\">|<\/span> <span style=\"color: #E06C75\">factorize<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">40<\/span><span style=\"color: #ABB2BF\">)))<\/span><\/code><\/pre>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\">{2: 2, 7: 2} {2: 2, 5: 1, 13: 1}\n\u041d\u041e\u0414: 4 4\n\u041d\u041e\u041a: 12740 12740\n\u041d\u041e\u0414\/\u041d\u041e\u041a 15 \u0438 40: 5 120\n<\/code><\/pre>\n<p>\u0420\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u0441\u043e\u0432\u043f\u0430\u0434\u0430\u0435\u0442 \u0441\u043e \u0441\u0442\u0430\u043d\u0434\u0430\u0440\u0442\u043d\u044b\u043c\u0438 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">math.gcd<\/code> \u0438 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">math.lcm<\/code> (lcm \u0434\u043e\u0441\u0442\u0443\u043f\u043d\u0430 \u0441 Python 3.9). \u0413\u0440\u0430\u043d\u0438\u0446\u0430 \u043c\u0435\u0442\u043e\u0434\u0430: \u0447\u0435\u0440\u0435\u0437 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0443\u0434\u043e\u0431\u043d\u043e \u0441\u0447\u0438\u0442\u0430\u0442\u044c \u0432\u0440\u0443\u0447\u043d\u0443\u044e \u0438 \u043e\u0431\u044a\u044f\u0441\u043d\u044f\u0442\u044c, \u043d\u043e \u0434\u043b\u044f \u0431\u043e\u043b\u044c\u0448\u0438\u0445 \u0447\u0438\u0441\u0435\u043b \u041d\u041e\u0414 \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u044e\u0442 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u043e\u043c \u0415\u0432\u043a\u043b\u0438\u0434\u0430, \u043a\u043e\u0442\u043e\u0440\u043e\u043c\u0443 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0435 \u043d\u0443\u0436\u043d\u043e, &#8212; \u043a\u0430\u043a \u0440\u0430\u0437 \u043f\u043e\u0442\u043e\u043c\u0443, \u0447\u0442\u043e \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0434\u043e\u0440\u043e\u0433\u043e\u0435.<\/p>\n<h2 id=\"s7\">\u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435 2: \u043f\u043e\u0447\u0435\u043c\u0443 \u043d\u0430 \u0442\u0440\u0443\u0434\u043d\u043e\u0441\u0442\u0438 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u044f \u0434\u0435\u0440\u0436\u0438\u0442\u0441\u044f RSA<\/h2>\n<p>\u0412 RSA \u043e\u0442\u043a\u0440\u044b\u0442\u044b\u0439 \u043a\u043b\u044e\u0447 &#8212; \u044d\u0442\u043e \u043c\u043e\u0434\u0443\u043b\u044c n = p * q (\u0434\u0432\u0430 \u0431\u043e\u043b\u044c\u0448\u0438\u0445 \u043f\u0440\u043e\u0441\u0442\u044b\u0445) \u0438 \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0442\u0430 e. \u0417\u0430\u043a\u0440\u044b\u0442\u0430\u044f \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0442\u0430 d \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u0435\u0442\u0441\u044f \u0438\u0437 \u03c6(n) = (p &#8212; 1) * (q &#8212; 1), \u0430 \u03c6(n) \u043b\u0435\u0433\u043a\u043e \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c, \u0442\u043e\u043b\u044c\u043a\u043e \u0437\u043d\u0430\u044f p \u0438 q. \u041f\u0435\u0440\u0435\u043c\u043d\u043e\u0436\u0438\u0442\u044c \u0434\u0432\u0430 \u043f\u0440\u043e\u0441\u0442\u044b\u0445 \u0431\u044b\u0441\u0442\u0440\u043e, \u0440\u0430\u0437\u043b\u043e\u0436\u0438\u0442\u044c \u0438\u0445 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043e\u0431\u0440\u0430\u0442\u043d\u043e \u043d\u0430 \u0441\u043e\u0432\u0440\u0435\u043c\u0435\u043d\u043d\u044b\u0445 \u0434\u043b\u0438\u043d\u0430\u0445 \u043a\u043b\u044e\u0447\u0430 \u0438\u0437\u0432\u0435\u0441\u0442\u043d\u044b\u043c\u0438 \u043a\u043b\u0430\u0441\u0441\u0438\u0447\u0435\u0441\u043a\u0438\u043c\u0438 \u043c\u0435\u0442\u043e\u0434\u0430\u043c\u0438 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u043d\u0435\u043b\u044c\u0437\u044f.<\/p>\n<p>\u0423\u0447\u0435\u0431\u043d\u0430\u044f \u043c\u043e\u0434\u0435\u043b\u044c \u043d\u0430 \u043a\u0440\u043e\u0448\u0435\u0447\u043d\u044b\u0445 \u0447\u0438\u0441\u043b\u0430\u0445 (\u043d\u0435 \u0448\u0438\u0444\u0440\u043e\u0432\u0430\u043d\u0438\u0435, \u0430 \u0434\u0435\u043c\u043e\u043d\u0441\u0442\u0440\u0430\u0446\u0438\u044f \u0438\u0434\u0435\u0438):<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #7F848E\"># \u0423\u0447\u0435\u0431\u043d\u0430\u044f RSA \u043d\u0430 \u043a\u0440\u043e\u0448\u0435\u0447\u043d\u044b\u0445 \u0447\u0438\u0441\u043b\u0430\u0445: \u041d\u0415 \u0434\u043b\u044f \u0440\u0435\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0448\u0438\u0444\u0440\u043e\u0432\u0430\u043d\u0438\u044f<\/span>\n<span style=\"color: #E06C75\">p<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">q<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #D19A66\">61<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">53<\/span>\n<span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">p<\/span> <span style=\"color: #56B6C2\">*<\/span> <span style=\"color: #E06C75\">q<\/span>                  <span style=\"color: #7F848E\"># \u043e\u0442\u043a\u0440\u044b\u0442\u044b\u0439 \u043c\u043e\u0434\u0443\u043b\u044c<\/span>\n<span style=\"color: #E06C75\">phi<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">p<\/span> <span style=\"color: #56B6C2\">-<\/span> <span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #56B6C2\">*<\/span> <span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">q<\/span> <span style=\"color: #56B6C2\">-<\/span> <span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">)<\/span>    <span style=\"color: #7F848E\"># \u0437\u043d\u0430\u0435\u0442 \u0442\u043e\u043b\u044c\u043a\u043e \u0442\u043e\u0442, \u043a\u0442\u043e \u0437\u043d\u0430\u0435\u0442 p \u0438 q<\/span>\n<span style=\"color: #E06C75\">e<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #D19A66\">17<\/span>                     <span style=\"color: #7F848E\"># \u043e\u0442\u043a\u0440\u044b\u0442\u0430\u044f \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0442\u0430, \u0432\u0437\u0430\u0438\u043c\u043d\u043e \u043f\u0440\u043e\u0441\u0442\u0430 \u0441 phi<\/span>\n<span style=\"color: #E06C75\">d<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E5C07B\">pow<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">e<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #56B6C2\">-<\/span><span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">phi<\/span><span style=\"color: #ABB2BF\">)<\/span>        <span style=\"color: #7F848E\"># \u0437\u0430\u043a\u0440\u044b\u0442\u0430\u044f \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0442\u0430 (Python 3.8+)<\/span>\n<span style=\"color: #E06C75\">m<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #D19A66\">65<\/span>\n<span style=\"color: #E06C75\">c<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E5C07B\">pow<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">m<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">e<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">)<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #98C379\">f\"n={<\/span><span style=\"color: #E06C75\">n<\/span><span style=\"color: #98C379\">} phi={<\/span><span style=\"color: #E06C75\">phi<\/span><span style=\"color: #98C379\">} d={<\/span><span style=\"color: #E06C75\">d<\/span><span style=\"color: #98C379\">}\"<\/span><span style=\"color: #ABB2BF\">)<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #98C379\">\"\u0448\u0438\u0444\u0440:\"<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">c<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #98C379\">\"\u0440\u0430\u0441\u0448\u0438\u0444\u0440\u043e\u0432\u043a\u0430:\"<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E5C07B\">pow<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">c<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">d<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">))<\/span>\n\n<span style=\"color: #7F848E\"># \u0410\u0442\u0430\u043a\u0443\u044e\u0449\u0438\u0439 \u0432\u0438\u0434\u0438\u0442 \u0442\u043e\u043b\u044c\u043a\u043e (n, e) - \u0438 \u0440\u0430\u0441\u043a\u043b\u0430\u0434\u044b\u0432\u0430\u0435\u0442 n \u043f\u0435\u0440\u0435\u0431\u043e\u0440\u043e\u043c<\/span>\n<span style=\"color: #E06C75\">f<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E5C07B\">next<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">x<\/span> <span style=\"color: #C678DD\">for<\/span> <span style=\"color: #E06C75\">x<\/span> <span style=\"color: #56B6C2\">in<\/span> <span style=\"color: #E5C07B\">range<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">2<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #C678DD\">if<\/span> <span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">%<\/span> <span style=\"color: #E06C75\">x<\/span> <span style=\"color: #56B6C2\">==<\/span> <span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">)<\/span>\n<span style=\"color: #E06C75\">d_found<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E5C07B\">pow<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">e<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #56B6C2\">-<\/span><span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">f<\/span> <span style=\"color: #56B6C2\">-<\/span> <span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #56B6C2\">*<\/span> <span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">\/\/<\/span> <span style=\"color: #E06C75\">f<\/span> <span style=\"color: #56B6C2\">-<\/span> <span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">))<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #98C379\">\"\u043d\u0430\u0439\u0434\u0435\u043d \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c:\"<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">f<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #98C379\">\"d \u0430\u0442\u0430\u043a\u0443\u044e\u0449\u0435\u0433\u043e:\"<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">d_found<\/span><span style=\"color: #ABB2BF\">)<\/span><\/code><\/pre>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\">n=3233 phi=3120 d=2753\n\u0448\u0438\u0444\u0440: 2790 \u0440\u0430\u0441\u0448\u0438\u0444\u0440\u043e\u0432\u043a\u0430: 65\n\u043d\u0430\u0439\u0434\u0435\u043d \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c: 53 d \u0430\u0442\u0430\u043a\u0443\u044e\u0449\u0435\u0433\u043e: 2753\n<\/code><\/pre>\n<p>\u041c\u043e\u0434\u0443\u043b\u044c 3233 \u0440\u0430\u0441\u043a\u043b\u0430\u0434\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u043c\u0433\u043d\u043e\u0432\u0435\u043d\u043d\u043e, \u0438 \u0430\u0442\u0430\u043a\u0443\u044e\u0449\u0438\u0439 \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u0442 \u0442\u043e\u0442 \u0436\u0435 \u0437\u0430\u043a\u0440\u044b\u0442\u044b\u0439 \u043a\u043b\u044e\u0447 2753. \u041f\u043e\u044d\u0442\u043e\u043c\u0443 \u0440\u0435\u0430\u043b\u044c\u043d\u044b\u0435 \u043c\u043e\u0434\u0443\u043b\u0438 \u0431\u0435\u0440\u0443\u0442 \u0434\u043b\u0438\u043d\u043e\u0439 \u043e\u0442 2048 \u0431\u0438\u0442: \u043f\u0435\u0440\u0435\u0431\u043e\u0440\u0443 \u0442\u0430\u043c \u043d\u0443\u0436\u043d\u043e \u043e\u043a\u043e\u043b\u043e 10^308 \u0434\u0435\u043b\u0435\u043d\u0438\u0439 (\u0441\u0442\u0440\u043e\u043a\u0430 \u0442\u0430\u0431\u043b\u0438\u0446\u044b \u0432\u044b\u0448\u0435).<\/p>\n<p>\u0413\u0440\u0430\u043d\u0438\u0446\u0430 \u044d\u0442\u043e\u0433\u043e \u043f\u0440\u0438\u043c\u0435\u0440\u0430: \u00ab\u0443\u0447\u0435\u0431\u043d\u0438\u043a RSA\u00bb \u0431\u0435\u0437 \u043f\u0430\u0434\u0434\u0438\u043d\u0433\u0430 \u043d\u0435\u0431\u0435\u0437\u043e\u043f\u0430\u0441\u0435\u043d \u0434\u0430\u0436\u0435 \u043f\u0440\u0438 \u0434\u043b\u0438\u043d\u043d\u043e\u043c \u043a\u043b\u044e\u0447\u0435 &#8212; \u043e\u0434\u0438\u043d\u0430\u043a\u043e\u0432\u044b\u0435 \u0441\u043e\u043e\u0431\u0449\u0435\u043d\u0438\u044f \u0434\u0430\u044e\u0442 \u043e\u0434\u0438\u043d\u0430\u043a\u043e\u0432\u044b\u0439 \u0448\u0438\u0444\u0440, \u0432\u043e\u0437\u043c\u043e\u0436\u043d\u044b \u0430\u0442\u0430\u043a\u0438 \u043d\u0430 \u043c\u0430\u043b\u044b\u0435 \u0441\u043e\u043e\u0431\u0449\u0435\u043d\u0438\u044f. \u0412 \u0440\u0435\u0430\u043b\u044c\u043d\u043e\u043c \u043a\u043e\u0434\u0435 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044e\u0442 \u043f\u0440\u043e\u0432\u0435\u0440\u0435\u043d\u043d\u044b\u0435 \u0431\u0438\u0431\u043b\u0438\u043e\u0442\u0435\u043a\u0438 \u0441\u043e \u0441\u0445\u0435\u043c\u0430\u043c\u0438 OAEP (\u0448\u0438\u0444\u0440\u043e\u0432\u0430\u043d\u0438\u0435) \u0438 PSS (\u043f\u043e\u0434\u043f\u0438\u0441\u044c) \u0438 \u043d\u0435 \u043f\u0438\u0448\u0443\u0442 RSA \u0432\u0440\u0443\u0447\u043d\u0443\u044e. \u041a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, \u0442\u0440\u0443\u0434\u043d\u043e\u0441\u0442\u044c \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u044f &#8212; \u043d\u0435 \u0434\u043e\u043a\u0430\u0437\u0430\u043d\u043d\u044b\u0439 \u0444\u0430\u043a\u0442, \u0430 \u043e\u0442\u0441\u0443\u0442\u0441\u0442\u0432\u0438\u0435 \u0438\u0437\u0432\u0435\u0441\u0442\u043d\u043e\u0433\u043e \u0431\u044b\u0441\u0442\u0440\u043e\u0433\u043e \u043a\u043b\u0430\u0441\u0441\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0430; \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u0428\u043e\u0440\u0430 \u0440\u0435\u0448\u0438\u043b \u0431\u044b \u0437\u0430\u0434\u0430\u0447\u0443 \u043d\u0430 \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e \u0431\u043e\u043b\u044c\u0448\u043e\u043c \u043a\u0432\u0430\u043d\u0442\u043e\u0432\u043e\u043c \u043a\u043e\u043c\u043f\u044c\u044e\u0442\u0435\u0440\u0435, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0440\u0430\u0437\u0432\u0438\u0432\u0430\u0435\u0442\u0441\u044f \u043f\u043e\u0441\u0442\u043a\u0432\u0430\u043d\u0442\u043e\u0432\u0430\u044f \u043a\u0440\u0438\u043f\u0442\u043e\u0433\u0440\u0430\u0444\u0438\u044f.<\/p>\n<h2 id=\"s8\">\u0412\u044b\u0432\u043e\u0434\u044b<\/h2>\n<ul>\n<li>\u0421\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c &#8212; \u043b\u044e\u0431\u043e\u0435 \u0447\u0438\u0441\u043b\u043e \u0432 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0438; \u043f\u0440\u043e\u0441\u0442\u043e\u0439 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c &#8212; \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043f\u0440\u043e\u0441\u0442\u044b\u043c; \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u0435\u0439 \u0443 \u0447\u0438\u0441\u043b\u0430 \u043e\u0431\u044b\u0447\u043d\u043e \u0431\u043e\u043b\u044c\u0448\u0435, \u0447\u0435\u043c \u043f\u0440\u043e\u0441\u0442\u044b\u0445 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0435\u0439.<\/li>\n<li>\u041e\u0441\u043d\u043e\u0432\u043d\u0430\u044f \u0442\u0435\u043e\u0440\u0435\u043c\u0430 \u0430\u0440\u0438\u0444\u043c\u0435\u0442\u0438\u043a\u0438 \u0433\u0430\u0440\u0430\u043d\u0442\u0438\u0440\u0443\u0435\u0442, \u0447\u0442\u043e \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0447\u0438\u0441\u043b\u0430 \u0431\u043e\u043b\u044c\u0448\u0435 1 \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u044b\u0435 \u0435\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u043e \u0441 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u044c\u044e \u0434\u043e \u043f\u043e\u0440\u044f\u0434\u043a\u0430; \u043a\u0430\u043d\u043e\u043d\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0437\u0430\u043f\u0438\u0441\u044c &#8212; \u0441\u0442\u0435\u043f\u0435\u043d\u0438 \u043f\u0440\u043e\u0441\u0442\u044b\u0445 \u043f\u043e \u0432\u043e\u0437\u0440\u0430\u0441\u0442\u0430\u043d\u0438\u044e.<\/li>\n<li>\u041f\u0435\u0440\u0435\u0431\u043e\u0440 \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u0435\u0439 \u0434\u043e \u221an \u0441 \u043f\u0440\u043e\u0432\u0435\u0440\u043a\u043e\u0439 \u043e\u0441\u0442\u0430\u0442\u043a\u0430 \u043a\u043e\u0440\u0440\u0435\u043a\u0442\u043d\u043e \u0440\u0430\u0441\u043a\u043b\u0430\u0434\u044b\u0432\u0430\u0435\u0442 \u043b\u044e\u0431\u043e\u0435 \u0447\u0438\u0441\u043b\u043e, \u043d\u043e \u0432 \u0445\u0443\u0434\u0448\u0435\u043c \u0441\u043b\u0443\u0447\u0430\u0435 \u0442\u0440\u0435\u0431\u0443\u0435\u0442 \u043e\u043a\u043e\u043b\u043e \u221an \/ 2 \u0434\u0435\u043b\u0435\u043d\u0438\u0439 &#8212; \u044d\u043a\u0441\u043f\u043e\u043d\u0435\u043d\u0446\u0438\u0430\u043b\u044c\u043d\u043e \u043e\u0442 \u0447\u0438\u0441\u043b\u0430 \u0446\u0438\u0444\u0440.<\/li>\n<li>\u0427\u0435\u0440\u0435\u0437 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u041d\u041e\u0414 \u0431\u0435\u0440\u0435\u0442 \u043c\u0438\u043d\u0438\u043c\u0443\u043c \u0441\u0442\u0435\u043f\u0435\u043d\u0435\u0439 \u043e\u0431\u0449\u0438\u0445 \u043f\u0440\u043e\u0441\u0442\u044b\u0445, \u041d\u041e\u041a &#8212; \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c \u0441\u0442\u0435\u043f\u0435\u043d\u0435\u0439 \u0432\u0441\u0435\u0445 \u043f\u0440\u043e\u0441\u0442\u044b\u0445; \u0434\u043b\u044f \u0431\u043e\u043b\u044c\u0448\u0438\u0445 \u0447\u0438\u0441\u0435\u043b \u041d\u041e\u0414 \u0441\u0447\u0438\u0442\u0430\u044e\u0442 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u043e\u043c \u0415\u0432\u043a\u043b\u0438\u0434\u0430.<\/li>\n<li>RSA \u043e\u043f\u0438\u0440\u0430\u0435\u0442\u0441\u044f \u043d\u0430 \u0442\u043e, \u0447\u0442\u043e \u0440\u0430\u0437\u043b\u043e\u0436\u0438\u0442\u044c \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0434\u0432\u0443\u0445 \u0431\u043e\u043b\u044c\u0448\u0438\u0445 \u043f\u0440\u043e\u0441\u0442\u044b\u0445 \u0438\u0437\u0432\u0435\u0441\u0442\u043d\u044b\u043c\u0438 \u043a\u043b\u0430\u0441\u0441\u0438\u0447\u0435\u0441\u043a\u0438\u043c\u0438 \u043c\u0435\u0442\u043e\u0434\u0430\u043c\u0438 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u043d\u0435\u0432\u043e\u0437\u043c\u043e\u0436\u043d\u043e.<\/li>\n<\/ul>\n<h2 id=\"s9\">\u0413\u0434\u0435 \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0435\u0442\u0441\u044f \/ \u0441\u0432\u044f\u0437\u044c \u0441 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u043e\u0439<\/h2>\n<div class=\"oj-cta\" style=\"background:#fbf4e4;border-left:4px solid #c9962b;border-radius:8px;padding:16px 20px;margin:24px 0\">\n<p style=\"margin:0 0 8px\"><strong>\u041e\u0441\u0432\u043e\u0439\u0442\u0435 \u0442\u0435\u043c\u0443 \u043d\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u0435<\/strong><\/p>\n<p>\u0420\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438 &#8212; \u043a\u043b\u0430\u0441\u0441\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0432\u0445\u043e\u0434 \u0432 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0438\u0447\u0435\u0441\u043a\u0443\u044e \u0442\u0435\u043e\u0440\u0438\u044e \u0447\u0438\u0441\u0435\u043b: \u043e\u0446\u0435\u043d\u043a\u0430 \u0441\u043b\u043e\u0436\u043d\u043e\u0441\u0442\u0438 \u043f\u043e \u0434\u043b\u0438\u043d\u0435 \u0432\u0445\u043e\u0434\u0430, \u0440\u0435\u0448\u0435\u0442\u043e \u042d\u0440\u0430\u0442\u043e\u0441\u0444\u0435\u043d\u0430, \u043c\u043e\u0434\u0443\u043b\u044c\u043d\u0430\u044f \u0430\u0440\u0438\u0444\u043c\u0435\u0442\u0438\u043a\u0430, \u0431\u044b\u0441\u0442\u0440\u043e\u0435 \u0432\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u044c, \u0442\u0435\u0441\u0442\u044b \u043f\u0440\u043e\u0441\u0442\u043e\u0442\u044b. \u042d\u0442\u0438 \u0442\u0435\u043c\u044b \u0440\u0435\u0433\u0443\u043b\u044f\u0440\u043d\u043e \u0432\u0441\u0442\u0440\u0435\u0447\u0430\u044e\u0442\u0441\u044f \u0432 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u043d\u044b\u0445 \u0437\u0430\u0434\u0430\u0447\u0430\u0445 \u0438 \u043d\u0430 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0441\u043e\u0431\u0435\u0441\u0435\u0434\u043e\u0432\u0430\u043d\u0438\u044f\u0445. \u0421\u0438\u0441\u0442\u0435\u043c\u043d\u043e \u0440\u0430\u0437\u043e\u0431\u0440\u0430\u0442\u044c \u0438\u0445 \u0441 \u043e\u0446\u0435\u043d\u043a\u043e\u0439 \u0441\u043b\u043e\u0436\u043d\u043e\u0441\u0442\u0438 \u0438 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u043e\u0439 \u043d\u0430 \u043a\u043e\u0434\u0435 \u043c\u043e\u0436\u043d\u043e \u043d\u0430 \u043a\u0443\u0440\u0441\u0435 <a href=\"https:\/\/otus.ru\/lessons\/algorithm\/?int_article=teoriya-chisel-i-ih-razlozhenie&amp;int_place=article&amp;int_variant=v1\">\u00ab\u0410\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u044b \u0438 \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u044b \u0434\u0430\u043d\u043d\u044b\u0445\u00bb<\/a>. \u041f\u043e\u043f\u0440\u043e\u0431\u043e\u0432\u0430\u0442\u044c \u0444\u043e\u0440\u043c\u0430\u0442 \u0438 \u0437\u0430\u0434\u0430\u0442\u044c \u0432\u043e\u043f\u0440\u043e\u0441\u044b \u043f\u0440\u0435\u043f\u043e\u0434\u0430\u0432\u0430\u0442\u0435\u043b\u044f\u043c &#8212; \u043d\u0430 <a href=\"https:\/\/otus.ru\/events\/?int_article=teoriya-chisel-i-ih-razlozhenie&amp;int_place=article&amp;int_variant=v1\">\u0431\u0435\u0441\u043f\u043b\u0430\u0442\u043d\u044b\u0445 \u043e\u0442\u043a\u0440\u044b\u0442\u044b\u0445 \u0443\u0440\u043e\u043a\u0430\u0445 Otus<\/a>.<\/p>\n<\/div>\n<h2 id=\"s10\">FAQ<\/h2>\n<p><strong>\u041c\u043e\u0436\u043d\u043e \u043b\u0438 \u0440\u0430\u0437\u043b\u043e\u0436\u0438\u0442\u044c \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u044b\u0435 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438 \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e \u0438\u043b\u0438 \u043d\u043e\u043b\u044c?<\/strong><br \/>\n\u041e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e \u0437\u0430\u043f\u0438\u0441\u044b\u0432\u0430\u044e\u0442 \u043a\u0430\u043a -1, \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u043d\u043e\u0435 \u043d\u0430 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u043c\u043e\u0434\u0443\u043b\u044f: -84 = -1 * 2^2 * 3 * 7. \u0423 \u043d\u0443\u043b\u044f \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u044f \u043d\u0435\u0442: \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0441\u0442\u044b\u0445 \u0447\u0438\u0441\u0435\u043b \u043d\u0438\u043a\u043e\u0433\u0434\u0430 \u043d\u0435 \u0440\u0430\u0432\u043d\u043e 0. \u0427\u0438\u0441\u043b\u043e 1 \u043f\u043e \u0441\u043e\u0433\u043b\u0430\u0448\u0435\u043d\u0438\u044e \u0441\u0447\u0438\u0442\u0430\u044e\u0442 \u043f\u0443\u0441\u0442\u044b\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435\u043c &#8212; \u0432 \u043d\u0435\u043c \u043d\u0435\u0442 \u043d\u0438 \u043e\u0434\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0441\u0442\u043e\u0433\u043e \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044f.<\/p>\n<p><strong>\u0427\u0435\u043c \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438 \u043e\u0442\u043b\u0438\u0447\u0430\u0435\u0442\u0441\u044f \u043e\u0442 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u044f \u043c\u043d\u043e\u0433\u043e\u0447\u043b\u0435\u043d\u0430?<\/strong><br \/>\n\u0418\u0434\u0435\u044f \u0442\u0430 \u0436\u0435 &#8212; \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0438\u0442\u044c \u043e\u0431\u044a\u0435\u043a\u0442 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435\u043c \u043d\u0435\u0440\u0430\u0437\u043b\u043e\u0436\u0438\u043c\u044b\u0445 \u0447\u0430\u0441\u0442\u0435\u0439, \u043d\u043e \u0434\u043b\u044f \u043c\u043d\u043e\u0433\u043e\u0447\u043b\u0435\u043d\u043e\u0432 \u043d\u0435\u0440\u0430\u0437\u043b\u043e\u0436\u0438\u043c\u043e\u0441\u0442\u044c \u0437\u0430\u0432\u0438\u0441\u0438\u0442 \u043e\u0442 \u043f\u043e\u043b\u044f \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442\u043e\u0432: x^2 + 1 \u043d\u0435 \u0440\u0430\u0441\u043a\u043b\u0430\u0434\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u043d\u0430\u0434 \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u044b\u043c\u0438 \u0447\u0438\u0441\u043b\u0430\u043c\u0438 \u0438 \u0440\u0430\u0441\u043a\u043b\u0430\u0434\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u043d\u0430\u0434 \u043a\u043e\u043c\u043f\u043b\u0435\u043a\u0441\u043d\u044b\u043c\u0438.<\/p>\n<p><strong>\u041a\u0430\u043a \u0431\u044b\u0441\u0442\u0440\u043e \u0440\u0430\u0437\u043b\u043e\u0436\u0438\u0442\u044c \u0441\u0440\u0430\u0437\u0443 \u0432\u0441\u0435 \u0447\u0438\u0441\u043b\u0430 \u0434\u043e N?<\/strong><br \/>\n\u0420\u0435\u0448\u0435\u0442\u043e\u043c \u0441 \u0441\u043e\u0445\u0440\u0430\u043d\u0435\u043d\u0438\u0435\u043c \u043d\u0430\u0438\u043c\u0435\u043d\u044c\u0448\u0435\u0433\u043e \u043f\u0440\u043e\u0441\u0442\u043e\u0433\u043e \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u044f \u0434\u043b\u044f \u043a\u0430\u0436\u0434\u043e\u0433\u043e \u0447\u0438\u0441\u043b\u0430: \u043f\u043e\u0441\u043b\u0435 \u043f\u0440\u0435\u0434\u043f\u043e\u0434\u0441\u0447\u0435\u0442\u0430 \u0437\u0430 \u0432\u0440\u0435\u043c\u044f \u043e\u043a\u043e\u043b\u043e N * log log N \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u043b\u044e\u0431\u043e\u0433\u043e \u0447\u0438\u0441\u043b\u0430 \u0434\u043e N \u0438\u0434\u0435\u0442 \u0434\u0435\u043b\u0435\u043d\u0438\u0435\u043c \u043d\u0430 \u0441\u043e\u0445\u0440\u0430\u043d\u0435\u043d\u043d\u044b\u0439 \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u044c \u0437\u0430 O(log n) \u0448\u0430\u0433\u043e\u0432.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0421\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c &#8212; \u044d\u0442\u043e \u043a\u0430\u0436\u0434\u043e\u0435 \u0438\u0437 \u0447\u0438\u0441\u0435\u043b, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043f\u0435\u0440\u0435\u043c\u043d\u043e\u0436\u0430\u044e\u0442 \u0432 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0438: \u0432 \u0437\u0430\u043f\u0438\u0441\u0438 2 * 3 * 7 = 42 \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438 &#8212; 2, 3 \u0438 7. \u0420\u0430\u0437\u043b\u043e\u0436\u0438\u0442\u044c \u0447\u0438\u0441\u043b\u043e \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u044b\u0435 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438 &#8212; \u0437\u043d\u0430\u0447\u0438\u0442 \u0437\u0430\u043f\u0438\u0441\u0430\u0442\u044c \u0435\u0433\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435\u043c \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0435\u0439, \u043a\u0430\u0436\u0434\u044b\u0439 \u0438\u0437 \u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u043f\u0440\u043e\u0441\u0442\u043e\u0435 \u0447\u0438\u0441\u043b\u043e. \u041d\u0438\u0436\u0435 &#8212; \u0442\u0435\u043e\u0440\u0435\u043c\u0430, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u0433\u0430\u0440\u0430\u043d\u0442\u0438\u0440\u0443\u0435\u0442, \u0447\u0442\u043e \u0442\u0430\u043a\u043e\u0435 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0435\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u043e, \u0440\u0443\u0447\u043d\u044b\u0435 \u0441\u043f\u043e\u0441\u043e\u0431\u044b, \u043a\u043e\u0434 \u043d\u0430 Python [&hellip;]<\/p>\n","protected":false},"author":11,"featured_media":5380,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[245],"class_list":["post-5379","post","type-post","status-publish","format-standard","has-post-thumbnail","","category-polza","tag-matematika-2"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.3 - aioseo.com -->\n\t<meta name=\"description\" content=\"\u0427\u0442\u043e \u0442\u0430\u043a\u043e\u0435 \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c, \u043e\u0441\u043d\u043e\u0432\u043d\u0430\u044f \u0442\u0435\u043e\u0440\u0435\u043c\u0430 \u0430\u0440\u0438\u0444\u043c\u0435\u0442\u0438\u043a\u0438 \u0438 \u043a\u0430\u043d\u043e\u043d\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0437\u0430\u043f\u0438\u0441\u044c \u0447\u0438\u0441\u043b\u0430. \u0420\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0441\u0442\u043e\u043b\u0431\u0438\u043a\u043e\u043c \u0438 \u0434\u0435\u0440\u0435\u0432\u043e\u043c, \u043a\u043e\u0434 \u043f\u0435\u0440\u0435\u0431\u043e\u0440\u043e\u043c \u0434\u043e \u043a\u043e\u0440\u043d\u044f, \u041d\u041e\u0414 \u0438 \u041d\u041e\u041a, \u0441\u0432\u044f\u0437\u044c \u0441 RSA.\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"A. Pavlenko\"\/>\n\t<link rel=\"canonical\" href=\"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/\" \/>\n\t\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 5.0.3\" \/>\n\t\t<script type=\"application\/ld+json\" class=\"aioseo-schema\">\n\t\t\t{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/teoriya-chisel-i-ih-razlozhenie\\\/#article\",\"name\":\"\\u0421\\u043e\\u043c\\u043d\\u043e\\u0436\\u0438\\u0442\\u0435\\u043b\\u044c \\u0438 \\u0440\\u0430\\u0437\\u043b\\u043e\\u0436\\u0435\\u043d\\u0438\\u0435 \\u0447\\u0438\\u0441\\u043b\\u0430 \\u043d\\u0430 \\u043f\\u0440\\u043e\\u0441\\u0442\\u044b\\u0435 \\u043c\\u043d\\u043e\\u0436\\u0438\\u0442\\u0435\\u043b\\u0438 | Otus\",\"headline\":\"\\u0420\\u0430\\u0437\\u043b\\u043e\\u0436\\u0435\\u043d\\u0438\\u0435 \\u0447\\u0438\\u0441\\u043b\\u0430 \\u043d\\u0430 \\u043f\\u0440\\u043e\\u0441\\u0442\\u044b\\u0435 \\u043c\\u043d\\u043e\\u0436\\u0438\\u0442\\u0435\\u043b\\u0438: \\u0441\\u043e\\u043c\\u043d\\u043e\\u0436\\u0438\\u0442\\u0435\\u043b\\u0438, \\u0430\\u043b\\u0433\\u043e\\u0440\\u0438\\u0442\\u043c \\u0438 \\u043a\\u043e\\u0434\",\"author\":{\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/author\\\/a-pavlenko\\\/#author\"},\"publisher\":{\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/#organization\"},\"image\":{\"@type\":\"ImageObject\",\"url\":\"https:\\\/\\\/otus.ru\\\/journal\\\/wp-content\\\/uploads\\\/2023\\\/02\\\/oj-1080x720-5.jpg\",\"width\":2245,\"height\":1587},\"datePublished\":\"2026-09-24T09:00:00+00:00\",\"dateModified\":\"2026-09-25T15:27:38+00:00\",\"inLanguage\":\"ru-RU\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/teoriya-chisel-i-ih-razlozhenie\\\/#webpage\"},\"isPartOf\":{\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/teoriya-chisel-i-ih-razlozhenie\\\/#webpage\"},\"articleSection\":\"\\u041f\\u043e\\u043b\\u0435\\u0437\\u043d\\u043e\\u0435, \\u043c\\u0430\\u0442\\u0435\\u043c\\u0430\\u0442\\u0438\\u043a\\u0430\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/teoriya-chisel-i-ih-razlozhenie\\\/#breadcrumblist\",\"itemListElement\":[{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/#listItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/otus.ru\\\/journal\\\/\",\"nextItem\":{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/category\\\/polza\\\/#listItem\",\"name\":\"\\u041f\\u043e\\u043b\\u0435\\u0437\\u043d\\u043e\\u0435\"}},{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/category\\\/polza\\\/#listItem\",\"position\":2,\"name\":\"\\u041f\\u043e\\u043b\\u0435\\u0437\\u043d\\u043e\\u0435\",\"item\":\"https:\\\/\\\/otus.ru\\\/journal\\\/category\\\/polza\\\/\",\"nextItem\":{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/teoriya-chisel-i-ih-razlozhenie\\\/#listItem\",\"name\":\"\\u0420\\u0430\\u0437\\u043b\\u043e\\u0436\\u0435\\u043d\\u0438\\u0435 \\u0447\\u0438\\u0441\\u043b\\u0430 \\u043d\\u0430 \\u043f\\u0440\\u043e\\u0441\\u0442\\u044b\\u0435 \\u043c\\u043d\\u043e\\u0436\\u0438\\u0442\\u0435\\u043b\\u0438: \\u0441\\u043e\\u043c\\u043d\\u043e\\u0436\\u0438\\u0442\\u0435\\u043b\\u0438, \\u0430\\u043b\\u0433\\u043e\\u0440\\u0438\\u0442\\u043c \\u0438 \\u043a\\u043e\\u0434\"},\"previousItem\":{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/#listItem\",\"name\":\"Home\"}},{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/teoriya-chisel-i-ih-razlozhenie\\\/#listItem\",\"position\":3,\"name\":\"\\u0420\\u0430\\u0437\\u043b\\u043e\\u0436\\u0435\\u043d\\u0438\\u0435 \\u0447\\u0438\\u0441\\u043b\\u0430 \\u043d\\u0430 \\u043f\\u0440\\u043e\\u0441\\u0442\\u044b\\u0435 \\u043c\\u043d\\u043e\\u0436\\u0438\\u0442\\u0435\\u043b\\u0438: \\u0441\\u043e\\u043c\\u043d\\u043e\\u0436\\u0438\\u0442\\u0435\\u043b\\u0438, \\u0430\\u043b\\u0433\\u043e\\u0440\\u0438\\u0442\\u043c \\u0438 \\u043a\\u043e\\u0434\",\"previousItem\":{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/category\\\/polza\\\/#listItem\",\"name\":\"\\u041f\\u043e\\u043b\\u0435\\u0437\\u043d\\u043e\\u0435\"}}]},{\"@type\":\"Organization\",\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/#organization\",\"name\":\"\\u041e\\u0442\\u0443\\u0441 \\u043e\\u043d\\u043b\\u0430\\u0439\\u043d-\\u043e\\u0431\\u0440\\u0430\\u0437\\u043e\\u0432\\u0430\\u043d\\u0438\\u0435\",\"description\":\"\\u0411\\u043b\\u043e\\u0433 \\u043f\\u0440\\u043e IT\",\"url\":\"https:\\\/\\\/otus.ru\\\/journal\\\/\",\"telephone\":\"+74999389202\",\"logo\":{\"@type\":\"ImageObject\",\"url\":\"https:\\\/\\\/otus.ru\\\/journal\\\/wp-content\\\/uploads\\\/2026\\\/04\\\/OTUS-journal-logo-180x180-1.png\",\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/teoriya-chisel-i-ih-razlozhenie\\\/#organizationLogo\",\"width\":180,\"height\":180},\"image\":{\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/teoriya-chisel-i-ih-razlozhenie\\\/#organizationLogo\"},\"sameAs\":[\"https:\\\/\\\/www.youtube.com\\\/channel\\\/UCetgtvy93o3i3CvyGXKFU3g\"]},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/author\\\/a-pavlenko\\\/#author\",\"url\":\"https:\\\/\\\/otus.ru\\\/journal\\\/author\\\/a-pavlenko\\\/\",\"name\":\"A. Pavlenko\",\"image\":{\"@type\":\"ImageObject\",\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/teoriya-chisel-i-ih-razlozhenie\\\/#authorImage\",\"url\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/13488c7d5b657aaaf1808afd398f71e0225df07dd94eb7e5fcdfc45feff43cb5?s=96&d=mm&r=g\",\"width\":96,\"height\":96,\"caption\":\"A. Pavlenko\"}},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/teoriya-chisel-i-ih-razlozhenie\\\/#webpage\",\"url\":\"https:\\\/\\\/otus.ru\\\/journal\\\/teoriya-chisel-i-ih-razlozhenie\\\/\",\"name\":\"\\u0421\\u043e\\u043c\\u043d\\u043e\\u0436\\u0438\\u0442\\u0435\\u043b\\u044c \\u0438 \\u0440\\u0430\\u0437\\u043b\\u043e\\u0436\\u0435\\u043d\\u0438\\u0435 \\u0447\\u0438\\u0441\\u043b\\u0430 \\u043d\\u0430 \\u043f\\u0440\\u043e\\u0441\\u0442\\u044b\\u0435 \\u043c\\u043d\\u043e\\u0436\\u0438\\u0442\\u0435\\u043b\\u0438 | Otus\",\"description\":\"\\u0427\\u0442\\u043e \\u0442\\u0430\\u043a\\u043e\\u0435 \\u0441\\u043e\\u043c\\u043d\\u043e\\u0436\\u0438\\u0442\\u0435\\u043b\\u044c, \\u043e\\u0441\\u043d\\u043e\\u0432\\u043d\\u0430\\u044f \\u0442\\u0435\\u043e\\u0440\\u0435\\u043c\\u0430 \\u0430\\u0440\\u0438\\u0444\\u043c\\u0435\\u0442\\u0438\\u043a\\u0438 \\u0438 \\u043a\\u0430\\u043d\\u043e\\u043d\\u0438\\u0447\\u0435\\u0441\\u043a\\u0430\\u044f \\u0437\\u0430\\u043f\\u0438\\u0441\\u044c \\u0447\\u0438\\u0441\\u043b\\u0430. \\u0420\\u0430\\u0437\\u043b\\u043e\\u0436\\u0435\\u043d\\u0438\\u0435 \\u0441\\u0442\\u043e\\u043b\\u0431\\u0438\\u043a\\u043e\\u043c \\u0438 \\u0434\\u0435\\u0440\\u0435\\u0432\\u043e\\u043c, \\u043a\\u043e\\u0434 \\u043f\\u0435\\u0440\\u0435\\u0431\\u043e\\u0440\\u043e\\u043c \\u0434\\u043e \\u043a\\u043e\\u0440\\u043d\\u044f, \\u041d\\u041e\\u0414 \\u0438 \\u041d\\u041e\\u041a, \\u0441\\u0432\\u044f\\u0437\\u044c \\u0441 RSA.\",\"inLanguage\":\"ru-RU\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/#website\"},\"breadcrumb\":{\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/teoriya-chisel-i-ih-razlozhenie\\\/#breadcrumblist\"},\"author\":{\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/author\\\/a-pavlenko\\\/#author\"},\"creator\":{\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/author\\\/a-pavlenko\\\/#author\"},\"image\":{\"@type\":\"ImageObject\",\"url\":\"https:\\\/\\\/otus.ru\\\/journal\\\/wp-content\\\/uploads\\\/2023\\\/02\\\/oj-1080x720-5.jpg\",\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/teoriya-chisel-i-ih-razlozhenie\\\/#mainImage\",\"width\":2245,\"height\":1587},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/teoriya-chisel-i-ih-razlozhenie\\\/#mainImage\"},\"datePublished\":\"2026-09-24T09:00:00+00:00\",\"dateModified\":\"2026-09-25T15:27:38+00:00\"},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/#website\",\"url\":\"https:\\\/\\\/otus.ru\\\/journal\\\/\",\"name\":\"OTUS \\u0416\\u0443\\u0440\\u043d\\u0430\\u043b - \\u0411\\u043b\\u043e\\u0433 \\u043f\\u0440\\u043e IT\",\"alternateName\":\"OTUS \\u0416\\u0443\\u0440\\u043d\\u0430\\u043b\",\"description\":\"\\u0411\\u043b\\u043e\\u0433 \\u043f\\u0440\\u043e IT\",\"inLanguage\":\"ru-RU\",\"publisher\":{\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/#organization\"}}]}\n\t\t<\/script>\n\t\t<!-- All in One SEO -->\n\n","aioseo_head_json":{"title":"\u0421\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c \u0438 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0447\u0438\u0441\u043b\u0430 \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u044b\u0435 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438 | Otus","description":"\u0427\u0442\u043e \u0442\u0430\u043a\u043e\u0435 \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c, \u043e\u0441\u043d\u043e\u0432\u043d\u0430\u044f \u0442\u0435\u043e\u0440\u0435\u043c\u0430 \u0430\u0440\u0438\u0444\u043c\u0435\u0442\u0438\u043a\u0438 \u0438 \u043a\u0430\u043d\u043e\u043d\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0437\u0430\u043f\u0438\u0441\u044c \u0447\u0438\u0441\u043b\u0430. \u0420\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0441\u0442\u043e\u043b\u0431\u0438\u043a\u043e\u043c \u0438 \u0434\u0435\u0440\u0435\u0432\u043e\u043c, \u043a\u043e\u0434 \u043f\u0435\u0440\u0435\u0431\u043e\u0440\u043e\u043c \u0434\u043e \u043a\u043e\u0440\u043d\u044f, \u041d\u041e\u0414 \u0438 \u041d\u041e\u041a, \u0441\u0432\u044f\u0437\u044c \u0441 RSA.","canonical_url":"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/","robots":"max-image-preview:large","keywords":"","webmasterTools":{"miscellaneous":""},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/#article","name":"\u0421\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c \u0438 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0447\u0438\u0441\u043b\u0430 \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u044b\u0435 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438 | Otus","headline":"\u0420\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0447\u0438\u0441\u043b\u0430 \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u044b\u0435 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438: \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438, \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u0438 \u043a\u043e\u0434","author":{"@id":"https:\/\/otus.ru\/journal\/author\/a-pavlenko\/#author"},"publisher":{"@id":"https:\/\/otus.ru\/journal\/#organization"},"image":{"@type":"ImageObject","url":"https:\/\/otus.ru\/journal\/wp-content\/uploads\/2023\/02\/oj-1080x720-5.jpg","width":2245,"height":1587},"datePublished":"2026-09-24T09:00:00+00:00","dateModified":"2026-09-25T15:27:38+00:00","inLanguage":"ru-RU","mainEntityOfPage":{"@id":"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/#webpage"},"isPartOf":{"@id":"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/#webpage"},"articleSection":"\u041f\u043e\u043b\u0435\u0437\u043d\u043e\u0435, \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430"},{"@type":"BreadcrumbList","@id":"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/#breadcrumblist","itemListElement":[{"@type":"ListItem","@id":"https:\/\/otus.ru\/journal\/#listItem","position":1,"name":"Home","item":"https:\/\/otus.ru\/journal\/","nextItem":{"@type":"ListItem","@id":"https:\/\/otus.ru\/journal\/category\/polza\/#listItem","name":"\u041f\u043e\u043b\u0435\u0437\u043d\u043e\u0435"}},{"@type":"ListItem","@id":"https:\/\/otus.ru\/journal\/category\/polza\/#listItem","position":2,"name":"\u041f\u043e\u043b\u0435\u0437\u043d\u043e\u0435","item":"https:\/\/otus.ru\/journal\/category\/polza\/","nextItem":{"@type":"ListItem","@id":"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/#listItem","name":"\u0420\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0447\u0438\u0441\u043b\u0430 \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u044b\u0435 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438: \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438, \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u0438 \u043a\u043e\u0434"},"previousItem":{"@type":"ListItem","@id":"https:\/\/otus.ru\/journal\/#listItem","name":"Home"}},{"@type":"ListItem","@id":"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/#listItem","position":3,"name":"\u0420\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0447\u0438\u0441\u043b\u0430 \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u044b\u0435 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438: \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438, \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u0438 \u043a\u043e\u0434","previousItem":{"@type":"ListItem","@id":"https:\/\/otus.ru\/journal\/category\/polza\/#listItem","name":"\u041f\u043e\u043b\u0435\u0437\u043d\u043e\u0435"}}]},{"@type":"Organization","@id":"https:\/\/otus.ru\/journal\/#organization","name":"\u041e\u0442\u0443\u0441 \u043e\u043d\u043b\u0430\u0439\u043d-\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435","description":"\u0411\u043b\u043e\u0433 \u043f\u0440\u043e IT","url":"https:\/\/otus.ru\/journal\/","telephone":"+74999389202","logo":{"@type":"ImageObject","url":"https:\/\/otus.ru\/journal\/wp-content\/uploads\/2026\/04\/OTUS-journal-logo-180x180-1.png","@id":"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/#organizationLogo","width":180,"height":180},"image":{"@id":"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/#organizationLogo"},"sameAs":["https:\/\/www.youtube.com\/channel\/UCetgtvy93o3i3CvyGXKFU3g"]},{"@type":"Person","@id":"https:\/\/otus.ru\/journal\/author\/a-pavlenko\/#author","url":"https:\/\/otus.ru\/journal\/author\/a-pavlenko\/","name":"A. Pavlenko","image":{"@type":"ImageObject","@id":"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/#authorImage","url":"https:\/\/secure.gravatar.com\/avatar\/13488c7d5b657aaaf1808afd398f71e0225df07dd94eb7e5fcdfc45feff43cb5?s=96&d=mm&r=g","width":96,"height":96,"caption":"A. Pavlenko"}},{"@type":"WebPage","@id":"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/#webpage","url":"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/","name":"\u0421\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c \u0438 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0447\u0438\u0441\u043b\u0430 \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u044b\u0435 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438 | Otus","description":"\u0427\u0442\u043e \u0442\u0430\u043a\u043e\u0435 \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c, \u043e\u0441\u043d\u043e\u0432\u043d\u0430\u044f \u0442\u0435\u043e\u0440\u0435\u043c\u0430 \u0430\u0440\u0438\u0444\u043c\u0435\u0442\u0438\u043a\u0438 \u0438 \u043a\u0430\u043d\u043e\u043d\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0437\u0430\u043f\u0438\u0441\u044c \u0447\u0438\u0441\u043b\u0430. \u0420\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0441\u0442\u043e\u043b\u0431\u0438\u043a\u043e\u043c \u0438 \u0434\u0435\u0440\u0435\u0432\u043e\u043c, \u043a\u043e\u0434 \u043f\u0435\u0440\u0435\u0431\u043e\u0440\u043e\u043c \u0434\u043e \u043a\u043e\u0440\u043d\u044f, \u041d\u041e\u0414 \u0438 \u041d\u041e\u041a, \u0441\u0432\u044f\u0437\u044c \u0441 RSA.","inLanguage":"ru-RU","isPartOf":{"@id":"https:\/\/otus.ru\/journal\/#website"},"breadcrumb":{"@id":"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/#breadcrumblist"},"author":{"@id":"https:\/\/otus.ru\/journal\/author\/a-pavlenko\/#author"},"creator":{"@id":"https:\/\/otus.ru\/journal\/author\/a-pavlenko\/#author"},"image":{"@type":"ImageObject","url":"https:\/\/otus.ru\/journal\/wp-content\/uploads\/2023\/02\/oj-1080x720-5.jpg","@id":"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/#mainImage","width":2245,"height":1587},"primaryImageOfPage":{"@id":"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/#mainImage"},"datePublished":"2026-09-24T09:00:00+00:00","dateModified":"2026-09-25T15:27:38+00:00"},{"@type":"WebSite","@id":"https:\/\/otus.ru\/journal\/#website","url":"https:\/\/otus.ru\/journal\/","name":"OTUS \u0416\u0443\u0440\u043d\u0430\u043b - \u0411\u043b\u043e\u0433 \u043f\u0440\u043e IT","alternateName":"OTUS \u0416\u0443\u0440\u043d\u0430\u043b","description":"\u0411\u043b\u043e\u0433 \u043f\u0440\u043e IT","inLanguage":"ru-RU","publisher":{"@id":"https:\/\/otus.ru\/journal\/#organization"}}]}},"aioseo_meta_data":{"post_id":"5379","title":"\u0421\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c \u0438 \u0440\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0447\u0438\u0441\u043b\u0430 \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u044b\u0435 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438 | Otus","description":"\u0427\u0442\u043e \u0442\u0430\u043a\u043e\u0435 \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c, \u043e\u0441\u043d\u043e\u0432\u043d\u0430\u044f \u0442\u0435\u043e\u0440\u0435\u043c\u0430 \u0430\u0440\u0438\u0444\u043c\u0435\u0442\u0438\u043a\u0438 \u0438 \u043a\u0430\u043d\u043e\u043d\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0437\u0430\u043f\u0438\u0441\u044c \u0447\u0438\u0441\u043b\u0430. \u0420\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0441\u0442\u043e\u043b\u0431\u0438\u043a\u043e\u043c \u0438 \u0434\u0435\u0440\u0435\u0432\u043e\u043c, \u043a\u043e\u0434 \u043f\u0435\u0440\u0435\u0431\u043e\u0440\u043e\u043c \u0434\u043e \u043a\u043e\u0440\u043d\u044f, \u041d\u041e\u0414 \u0438 \u041d\u041e\u041a, \u0441\u0432\u044f\u0437\u044c \u0441 RSA.","keywords":[],"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":[],"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"Article","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","location":null,"local_seo":null,"breadcrumb_settings":null,"limit_modified_date":false,"ai":null,"created":"2023-02-04 20:23:39","updated":"2026-09-25 17:49:49","seo_analyzer_scan_date":null,"focus_keyword":null,"additional_keywords":null,"truseo_locale":null},"aioseo_breadcrumb":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/otus.ru\/journal\/\" title=\"Home\">Home<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">\u00bb<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/otus.ru\/journal\/category\/polza\/\" title=\"\u041f\u043e\u043b\u0435\u0437\u043d\u043e\u0435\">\u041f\u043e\u043b\u0435\u0437\u043d\u043e\u0435<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">\u00bb<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\t\u0420\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0447\u0438\u0441\u043b\u0430 \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u044b\u0435 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438: \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438, \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u0438 \u043a\u043e\u0434\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"https:\/\/otus.ru\/journal\/"},{"label":"\u041f\u043e\u043b\u0435\u0437\u043d\u043e\u0435","link":"https:\/\/otus.ru\/journal\/category\/polza\/"},{"label":"\u0420\u0430\u0437\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0447\u0438\u0441\u043b\u0430 \u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u044b\u0435 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438: \u0441\u043e\u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438, \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u0438 \u043a\u043e\u0434","link":"https:\/\/otus.ru\/journal\/teoriya-chisel-i-ih-razlozhenie\/"}],"_links":{"self":[{"href":"https:\/\/otus.ru\/journal\/wp-json\/wp\/v2\/posts\/5379","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/otus.ru\/journal\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/otus.ru\/journal\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/otus.ru\/journal\/wp-json\/wp\/v2\/users\/11"}],"replies":[{"embeddable":true,"href":"https:\/\/otus.ru\/journal\/wp-json\/wp\/v2\/comments?post=5379"}],"version-history":[{"count":3,"href":"https:\/\/otus.ru\/journal\/wp-json\/wp\/v2\/posts\/5379\/revisions"}],"predecessor-version":[{"id":12910,"href":"https:\/\/otus.ru\/journal\/wp-json\/wp\/v2\/posts\/5379\/revisions\/12910"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/otus.ru\/journal\/wp-json\/wp\/v2\/media\/5380"}],"wp:attachment":[{"href":"https:\/\/otus.ru\/journal\/wp-json\/wp\/v2\/media?parent=5379"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/otus.ru\/journal\/wp-json\/wp\/v2\/categories?post=5379"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/otus.ru\/journal\/wp-json\/wp\/v2\/tags?post=5379"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}