{"id":35683,"date":"2025-12-26T18:56:30","date_gmt":"2025-12-26T18:56:30","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35683"},"modified":"2025-12-26T18:56:30","modified_gmt":"2025-12-26T18:56:30","slug":"%d1%87%d1%82%d0%be-%d1%82%d0%b0%d0%ba%d0%be%d0%b5-%d0%b4%d0%b5%d0%bb%d0%b8%d0%bc%d0%be%d1%81%d1%82%d1%8c","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/ru\/%d1%87%d1%82%d0%be-%d1%82%d0%b0%d0%ba%d0%be%d0%b5-%d0%b4%d0%b5%d0%bb%d0%b8%d0%bc%d0%be%d1%81%d1%82%d1%8c\/","title":{"rendered":"\u0427\u0442\u043e \u0442\u0430\u043a\u043e\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u044c?"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>\u0414\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u044c<\/h1>\n<p style=\"text-align:center;\"><em><br \/>\n<strong>\u0414\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u044c<\/strong> \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0440\u0435\u0430\u043b\u044c\u043d\u043e\u0439 \u043e\u0442\u043f\u0440\u0430\u0432\u043d\u043e\u0439 \u0442\u043e\u0447\u043a\u043e\u0439 \u0442\u0435\u043e\u0440\u0438\u0438 \u0447\u0438\u0441\u0435\u043b, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u043e\u043d\u0430 \u043f\u0440\u0435\u0432\u0440\u0430\u0449\u0430\u0435\u0442 \u0446\u0435\u043b\u044b\u0435 \u0447\u0438\u0441\u043b\u0430 \u0432 \u0441\u0438\u0441\u0442\u0435\u043c\u0443 \u0441\u043e \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u043e\u0439: \u0432\u044b \u043f\u0435\u0440\u0435\u0441\u0442\u0430\u0435\u0442\u0435 \u0440\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u0442\u044c \u0447\u0438\u0441\u043b\u0430 \u043a\u0430\u043a \u00ab\u043a\u043e\u043b\u0438\u0447\u0435\u0441\u0442\u0432\u0430\u00bb \u0438 \u043d\u0430\u0447\u0438\u043d\u0430\u0435\u0442\u0435 \u0432\u0438\u0434\u0435\u0442\u044c \u0438\u0445 \u043a\u0430\u043a \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u044b, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043b\u0438\u0431\u043e \u0441\u043e\u0433\u043b\u0430\u0441\u0443\u044e\u0442\u0441\u044f, \u043b\u0438\u0431\u043e \u043d\u0435 \u0441\u043e\u0433\u043b\u0430\u0441\u0443\u044e\u0442\u0441\u044f \u0434\u0440\u0443\u0433 \u0441 \u0434\u0440\u0443\u0433\u043e\u043c. \u0421 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u043e\u0434\u043d\u043e\u0439-\u0435\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0439 \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u044b, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, \u043c\u043e\u0436\u043d\u043e \u0432\u044b\u0440\u0430\u0437\u0438\u0442\u044c \u043a\u0430\u043a \u043a\u0440\u0438\u0442\u0435\u0440\u0438\u0438 \u0443\u043f\u0440\u043e\u0449\u0435\u043d\u0438\u044f \u0438 \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0437\u0430\u0446\u0438\u0438, \u0442\u0430\u043a \u0438 \u0441\u0443\u0442\u044c \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u043f\u0440\u043e\u0446\u0435\u0434\u0443\u0440, \u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0430 \u0415\u0432\u043a\u043b\u0438\u0434\u0430, \u043f\u043e\u0437\u0432\u043e\u043b\u044f\u044e\u0449\u0435\u0433\u043e \u0437\u0430 \u0441\u0447\u0438\u0442\u0430\u043d\u043d\u044b\u0435 \u0441\u0435\u043a\u0443\u043d\u0434\u044b \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u0442\u044c \u043d\u0430\u0438\u0431\u043e\u043b\u044c\u0448\u0438\u0435 \u043e\u0431\u0449\u0438\u0435 \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u0438 \u0434\u0430\u0436\u0435 \u0434\u043b\u044f \u0431\u043e\u043b\u044c\u0448\u0438\u0445 \u0447\u0438\u0441\u0435\u043b. \u041a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u044c \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0442\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u043e\u0441\u043d\u043e\u0432\u043e\u0439 \u0438\u0434\u0435\u0439, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0432\u043d\u043e\u0432\u044c \u0438 \u0432\u043d\u043e\u0432\u044c \u0432\u043e\u0437\u043d\u0438\u043a\u0430\u044e\u0442 \u0432 \u043f\u0440\u0438\u043a\u043b\u0430\u0434\u043d\u043e\u0439 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0435 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0435: \u0441\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f, \u043c\u043e\u0434\u0443\u043b\u044c\u043d\u0430\u044f \u0430\u0440\u0438\u0444\u043c\u0435\u0442\u0438\u043a\u0430, \u043f\u0440\u043e\u0432\u0435\u0440\u043a\u0438, \u043a\u043e\u0434\u044b \u0438 (\u043d\u0430 \u0431\u043e\u043b\u0435\u0435 \u043f\u043e\u0437\u0434\u043d\u0435\u043c \u044d\u0442\u0430\u043f\u0435) \u043a\u0440\u0438\u043f\u0442\u043e\u0433\u0440\u0430\u0444\u0438\u044f. \u041e\u0432\u043b\u0430\u0434\u0435\u0442\u044c \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u044c\u044e \u043f\u043e \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443 \u043e\u0437\u043d\u0430\u0447\u0430\u0435\u0442 \u043d\u0430\u0443\u0447\u0438\u0442\u044c\u0441\u044f \u043e\u0431\u043d\u0430\u0440\u0443\u0436\u0438\u0432\u0430\u0442\u044c \u043d\u0435\u0432\u0438\u0434\u0438\u043c\u044b\u0435 \u0437\u0430\u043a\u043e\u043d\u043e\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u0438 \u0432 \u0446\u0435\u043b\u044b\u0445 \u0447\u0438\u0441\u043b\u0430\u0445 \u0438 \u043f\u0440\u0435\u0432\u0440\u0430\u0449\u0430\u0442\u044c \u0438\u0445 \u0432 \u043f\u0440\u043e\u0446\u0435\u0434\u0443\u0440\u044b, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0440\u0430\u0431\u043e\u0442\u0430\u044e\u0442 \u0432\u0441\u0435\u0433\u0434\u0430.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><b>\u0426\u0435\u043b\u0438 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u044f<\/b><br \/>\n\u041f\u043e \u0437\u0430\u0432\u0435\u0440\u0448\u0435\u043d\u0438\u0438 \u0434\u0430\u043d\u043d\u043e\u0433\u043e \u043a\u043e\u043d\u0441\u043f\u0435\u043a\u0442\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442 \u0441\u043c\u043e\u0436\u0435\u0442:\n<\/p>\n<ol>\n<li><strong>\u041f\u043e\u043d\u0438\u043c\u0430\u0442\u044c<\/strong> \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438 \u043c\u0435\u0436\u0434\u0443 \u0446\u0435\u043b\u044b\u043c\u0438 \u0447\u0438\u0441\u043b\u0430\u043c\u0438.<\/li>\n<li><strong>\u041f\u043e\u043d\u0438\u043c\u0430\u0442\u044c<\/strong> \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438 \u0438 \u0435\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430.<\/li>\n<li><strong>\u0420\u0430\u0437\u0440\u0430\u0431\u0430\u0442\u044b\u0432\u0430\u0442\u044c<\/strong> \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u0430 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u043e\u0432 \u0438 \u0442\u0435\u043e\u0440\u0435\u043c, \u0441\u0432\u044f\u0437\u0430\u043d\u043d\u044b\u0445 \u0441 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u044c\u044e.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><b><u>\u041e\u0413\u041b\u0410\u0412\u041b\u0415\u041d\u0418\u0415<\/u><\/b><br \/>\n<a href=\"#1\">\u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438<\/a><br \/>\n<a href=\"#2\">\u041e\u0441\u043d\u043e\u0432\u043d\u044b\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438<\/a><br \/>\n<a href=\"#3\">\u041f\u0440\u0435\u0434\u043b\u0430\u0433\u0430\u0435\u043c\u044b\u0435 \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f<\/a>\n<\/p>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/HLrwdLse18U?si=tDiiV02P7ppdb4xF\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><br \/>\n<\/center><br \/>\n<a name=\"1\"><\/a><\/br><\/p>\n<h2>\u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438<\/h2>\n<p style=\"text-align: justify;\">\n\u041d\u0435\u0444\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u0430\u044f \u0438\u0434\u0435\u044f \u00ab<em>a \u0434\u0435\u043b\u0438\u0442 b<\/em>\u00bb \u0441\u0442\u0430\u043d\u043e\u0432\u0438\u0442\u0441\u044f \u0441\u0442\u0440\u043e\u0433\u043e\u0439, \u043a\u043e\u0433\u0434\u0430 \u043c\u044b \u0432\u044b\u0440\u0430\u0436\u0430\u0435\u043c \u0435\u0435 \u043a\u0430\u043a \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 \u0446\u0435\u043b\u044b\u043c\u0438 \u0447\u0438\u0441\u043b\u0430\u043c\u0438. \u0411\u0443\u0434\u0435\u043c \u0433\u043e\u0432\u043e\u0440\u0438\u0442\u044c, \u0447\u0442\u043e \u0446\u0435\u043b\u043e\u0435 \u0447\u0438\u0441\u043b\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> <strong>\u0434\u0435\u043b\u0438\u0442<\/strong> \u0446\u0435\u043b\u043e\u0435 \u0447\u0438\u0441\u043b\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span>, \u0435\u0441\u043b\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u043c\u043e\u0436\u0435\u0442 \u0431\u044b\u0442\u044c \u0437\u0430\u043f\u0438\u0441\u0430\u043d\u043e \u043a\u0430\u043a \u0442\u043e\u0447\u043d\u044b\u0439 \u043a\u0440\u0430\u0442\u043d\u044b\u0439 \u0447\u0438\u0441\u043b\u0430 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>. \u042d\u0442\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043e\u0441\u043d\u043e\u0432\u043e\u0439 \u0432\u0441\u0435\u0433\u043e \u043a\u043e\u043d\u0441\u043f\u0435\u043a\u0442\u0430, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u043e\u043d\u043e \u043f\u0440\u0435\u0432\u0440\u0430\u0449\u0430\u0435\u0442 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u0442\u0438\u043f\u0430 \u00ab\u0434\u0435\u043b\u0438\u0442\u0441\u044f \u0431\u0435\u0437 \u043e\u0441\u0442\u0430\u0442\u043a\u0430\u00bb \u0432 \u043f\u0440\u043e\u0432\u0435\u0440\u044f\u0435\u043c\u044b\u0439 \u043a\u0440\u0438\u0442\u0435\u0440\u0438\u0439.\n<\/p>\n<p style=\"text-align: justify;\">\n<strong>\u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435.<\/strong> \u041f\u0443\u0441\u0442\u044c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}<\/span><\/span> \u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>. \u0413\u043e\u0432\u043e\u0440\u044f\u0442, \u0447\u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u0434\u0435\u043b\u0438\u0442 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span>, \u0438 \u043f\u0438\u0448\u0443\u0442 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, \u0442\u043e\u0433\u0434\u0430 \u0438 \u0442\u043e\u043b\u044c\u043a\u043e \u0442\u043e\u0433\u0434\u0430, \u043a\u043e\u0433\u0434\u0430 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u0446\u0435\u043b\u043e\u0435 \u0447\u0438\u0441\u043b\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> \u0442\u0430\u043a\u043e\u0435, \u0447\u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=ka<\/span><\/span>. \u0412 \u043f\u0440\u043e\u0442\u0438\u0432\u043d\u043e\u043c \u0441\u043b\u0443\u0447\u0430\u0435 \u043f\u0438\u0448\u0443\u0442 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\nmid b<\/span><\/span>.\n<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b := (\\exists k \\in \\mathbb{Z})(b = ka )<\/span>\n<p style=\"text-align: justify;\">\n\u0412 \u044d\u0442\u043e\u043c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0438 \u0447\u0438\u0441\u043b\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f <strong>\u0447\u0430\u0441\u0442\u043d\u044b\u043c<\/strong> (\u0438\u043b\u0438 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0435\u043c), \u0441\u0432\u044f\u0437\u0430\u043d\u043d\u044b\u043c \u0441 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u044c\u044e. \u041d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u0443\u0442\u0432\u0435\u0440\u0436\u0434\u0435\u043d\u0438\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">6\\mid 42<\/span><\/span> \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u043d\u043e \u0443\u0442\u0432\u0435\u0440\u0436\u0434\u0435\u043d\u0438\u044e \u043e \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u043e\u0432\u0430\u043d\u0438\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> \u0442\u0430\u043a\u043e\u0433\u043e, \u0447\u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">42=6k<\/span><\/span>; \u0432 \u0434\u0430\u043d\u043d\u043e\u043c \u0441\u043b\u0443\u0447\u0430\u0435 \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e \u0432\u0437\u044f\u0442\u044c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=7<\/span><\/span>.\n<\/p>\n<h3>\u0412\u0430\u0436\u043d\u043e \u0443\u0447\u0438\u0442\u044b\u0432\u0430\u0442\u044c<\/h3>\n<ul>\n<li>\n    \u0423\u0441\u043b\u043e\u0432\u0438\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span> \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u043c. \u0414\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u043e, \u0435\u0441\u043b\u0438 \u0431\u044b \u043c\u044b \u043f\u043e\u043f\u044b\u0442\u0430\u043b\u0438\u0441\u044c \u0434\u043e\u043f\u0443\u0441\u0442\u0438\u0442\u044c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=0<\/span><\/span>, \u0442\u043e \u0443\u0441\u043b\u043e\u0432\u0438\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438 \u0442\u0440\u0435\u0431\u043e\u0432\u0430\u043b\u043e \u0431\u044b \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u043e\u0432\u0430\u043d\u0438\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> \u0442\u0430\u043a\u043e\u0433\u043e, \u0447\u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0\\cdot k<\/span><\/span>. \u041d\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\cdot k=0<\/span><\/span> \u0434\u043b\u044f \u043b\u044e\u0431\u043e\u0433\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span>, \u0442\u0430\u043a \u0447\u0442\u043e \u0435\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0439 \u0432\u043e\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442\u044c\u044e \u0431\u044b\u043b\u043e \u0431\u044b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span>. \u0412 \u044d\u0442\u043e\u043c \u0441\u043b\u0443\u0447\u0430\u0435 \u043d\u0435 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u043e\u0432\u0430\u043b\u043e \u0431\u044b \u00ab\u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u043e\u0433\u043e\u00bb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span>, \u0437\u0430\u0434\u0430\u0432\u0430\u0435\u043c\u043e\u0433\u043e \u0434\u0430\u043d\u043d\u044b\u043c \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435\u043c, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u043b\u044e\u0431\u043e\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> \u0443\u0434\u043e\u0432\u043b\u0435\u0442\u0432\u043e\u0440\u044f\u0435\u0442 \u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u0443 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=0\\cdot k<\/span><\/span>. \u0418\u043d\u044b\u043c\u0438 \u0441\u043b\u043e\u0432\u0430\u043c\u0438, \u043d\u0435\u0444\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0435 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=b\/a<\/span><\/span> \u043f\u0440\u0435\u0432\u0440\u0430\u0449\u0430\u0435\u0442\u0441\u044f \u0432 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0\/0<\/span><\/span>, \u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u043d\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043e. \u0427\u0442\u043e\u0431\u044b \u0438\u0437\u0431\u0435\u0436\u0430\u0442\u044c \u044d\u0442\u043e\u0439 \u0434\u0435\u0433\u0435\u043d\u0435\u0440\u0430\u0446\u0438\u0438 (\u043a\u043e\u0433\u0434\u0430 \u043f\u043e\u043d\u044f\u0442\u0438\u0435 \u0447\u0430\u0441\u0442\u043d\u043e\u0433\u043e \u0442\u0435\u0440\u044f\u0435\u0442 \u0441\u043c\u044b\u0441\u043b), \u0442\u0440\u0435\u0431\u0443\u0435\u0442\u0441\u044f \u0432\u044b\u043f\u043e\u043b\u043d\u0435\u043d\u0438\u0435 \u0443\u0441\u043b\u043e\u0432\u0438\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>. \u041f\u043e \u044d\u0442\u043e\u0439 \u043f\u0440\u0438\u0447\u0438\u043d\u0435 \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\mid b<\/span><\/span> \u043d\u0435 \u0441\u0447\u0438\u0442\u0430\u0435\u0442\u0441\u044f \u0434\u043e\u043f\u0443\u0441\u0442\u0438\u043c\u044b\u043c.\n<\/li>\n<li>\n        \u041d\u0430\u043f\u0440\u043e\u0442\u0438\u0432, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid 0<\/span><\/span> \u0432\u0435\u0440\u043d\u043e \u0434\u043b\u044f \u043b\u044e\u0431\u043e\u0433\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{Z}<\/span><\/span> \u043f\u0440\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e \u0432\u0437\u044f\u0442\u044c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>, \u0438 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f \u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=a\\cdot 0<\/span><\/span>.\n    <\/li>\n<\/ul>\n<p style=\"text-align: justify;\">\n\u0418\u0437 \u044d\u0442\u043e\u0433\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0441\u043b\u0435\u0434\u0443\u0435\u0442 \u044d\u043a\u0432\u0438\u0432\u0430\u043b\u0435\u043d\u0442\u043d\u043e\u0441\u0442\u044c, \u043a\u043e\u0442\u043e\u0440\u0443\u044e \u043c\u044b \u0431\u0443\u0434\u0435\u043c \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c \u043d\u0435\u043e\u0434\u043d\u043e\u043a\u0440\u0430\u0442\u043d\u043e: \u0443\u0442\u0432\u0435\u0440\u0436\u0434\u0435\u043d\u0438\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> \u0440\u0430\u0432\u043d\u043e\u0441\u0438\u043b\u044c\u043d\u043e \u0443\u0442\u0432\u0435\u0440\u0436\u0434\u0435\u043d\u0438\u044e, \u0447\u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u043f\u0440\u0438\u043d\u0430\u0434\u043b\u0435\u0436\u0438\u0442 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0443 \u0446\u0435\u043b\u044b\u0445 \u043a\u0440\u0430\u0442\u043d\u044b\u0445 \u0447\u0438\u0441\u043b\u0430 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>, \u0442\u043e \u0435\u0441\u0442\u044c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\in a\\mathbb{Z}<\/span><\/span>, \u0433\u0434\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mathbb{Z}=\\{ak:\\,k\\in\\mathbb{Z}\\}<\/span><\/span>. \u0422\u0430\u043a\u0430\u044f \u0437\u0430\u043f\u0438\u0441\u044c \u043f\u043e\u0434\u0447\u0435\u0440\u043a\u0438\u0432\u0430\u0435\u0442, \u0447\u0442\u043e \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u044c \u043d\u0435 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u00ab\u043f\u0440\u0438\u0435\u043c\u043e\u043c\u00bb, \u0430 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u0442 \u0441\u043e\u0431\u043e\u0439 \u0441\u043f\u043e\u0441\u043e\u0431 \u043e\u043f\u0438\u0441\u0430\u043d\u0438\u044f \u0441\u0442\u0440\u043e\u0433\u043e \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u044b\u0445 \u043f\u043e\u0434\u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432 \u0432\u043d\u0443\u0442\u0440\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{Z}<\/span><\/span>.\n<\/p>\n<p><a name=\"2\"><\/a><\/br><\/p>\n<h2>\u041e\u0441\u043d\u043e\u0432\u043d\u044b\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438<\/h2>\n<ul>\n<li><strong>\u0420\u0435\u0444\u043b\u0435\u043a\u0441\u0438\u0432\u043d\u043e\u0441\u0442\u044c:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid a<\/span><\/span>.<br \/>\n<u>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e<\/u>:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1)&amp;\\vdash a=ka \\leftrightarrow k=1 &amp;\\text{; \u041c\u0443\u043b\u044c\u0442\u0438\u043f\u043b\u0438\u043a\u0430\u0442\u0438\u0432\u043d\u0430\u044f \u0435\u0434\u0438\u043d\u0438\u0446\u0430 \u0432 $\\mathbb{Z}$}\\\\\n\n(2)&amp;\\vdash(\\exists k \\in \\mathbb{Z})(a=ka) &amp;\\text{; \u0418\u043d\u0442\u0440. \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u043e\u0432\u0430\u043d\u0438\u044f (1)}\\\\\n\n(3) &amp;\\vdash a \\mid a &amp;\\text{; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438 (2)} \\\\\n\n&amp;\\blacksquare &amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<li><strong>\u0422\u0440\u0430\u043d\u0437\u0438\u0442\u0438\u0432\u043d\u043e\u0441\u0442\u044c:<\/strong> \u0435\u0441\u043b\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> \u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\mid c<\/span><\/span>, \u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>.\n<p><u>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e<\/u>:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_1\\in\\mathbb{Z})(b=k_1a)  &amp;\\text{; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438, \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435}\\\\\n\n(2)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_2\\in\\mathbb{Z})(c=k_2b)  &amp;\\text{; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438, \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435}\\\\\n\n(3)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_1,k_2\\in\\mathbb{Z})(b=k_1a \\wedge c=k_2b)  &amp;\\text{; $\\exists$-\u043a\u043e\u043c\u043f\u0430\u043a\u0442\u0430\u0446\u0438\u044f (1,2)}\\\\\n\n(4)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_1,k_2\\in\\mathbb{Z})(k_2b=k_1k_2a \\wedge c=k_2b)  &amp;\\text{; \u0418\u0437 (3)}\\\\\n\n(5)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_1,k_2\\in\\mathbb{Z})( c=k_1k_2a)  &amp;\\text{; \u0418\u0437 (4)}\\\\\n\n&amp;\\text{\u0410\u043b\u0433\u0435\u0431\u0440\u0430 \u0432\u043d\u0443\u0442\u0440\u0438 \u043a\u0432\u0430\u043d\u0442\u0438\u0444\u0438\u043a\u0430\u0442\u043e\u0440\u0430}&amp; \\\\\n\n(6)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k\\in\\mathbb{Z})( c=ka)  &amp;\\text{; \u0418\u0437 (5)}\\\\\n\n&amp;\\text{\u0417\u0430\u043c\u043a\u043d\u0443\u0442\u043e\u0441\u0442\u044c $\\mathbb{Z}$ \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u043e \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u044f}&amp; \\\\\n\n(7)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash a\\mid c  &amp;\\text{; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438 (6)}\\\\\n\n(8)&amp; \\vdash (a\\mid b \\wedge  b\\mid c) \\rightarrow  a\\mid c  &amp;\\text{; $\\wedge$-TD (7)}\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<li><strong>\u0421\u043e\u0432\u043c\u0435\u0441\u0442\u0438\u043c\u043e\u0441\u0442\u044c \u0441\u043e \u0441\u043b\u043e\u0436\u0435\u043d\u0438\u0435\u043c \u0438 \u0432\u044b\u0447\u0438\u0442\u0430\u043d\u0438\u0435\u043c:<\/strong> \u0435\u0441\u043b\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> \u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>, \u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (b+c)<\/span><\/span> \u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (b-c)<\/span><\/span>.<br \/>\n<u>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e<\/u>:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_1 \\in \\mathbb{Z})(b=k_1 a) &amp;\\text{; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438, \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435}\\\\\n\n(2)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_2 \\in \\mathbb{Z})(c=k_2 a) &amp;\\text{; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438, \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435}\\\\\n\n(3)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_1, k_2 \\in \\mathbb{Z})(b=k_1 a \\wedge c=k_2 a) &amp;\\text{; $\\exists$-\u043a\u043e\u043c\u043f\u0430\u043a\u0442\u0430\u0446\u0438\u044f (1,2)}\\\\\n\n(4)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_1, k_2 \\in \\mathbb{Z})(b+c= (k_1+k_2)a) &amp;\\text{; \u0418\u0437 (3)}\\\\\n\n&amp;\\text{\u0410\u043b\u0433\u0435\u0431\u0440\u0430 \u0432\u043d\u0443\u0442\u0440\u0438 \u043a\u0432\u0430\u043d\u0442\u0438\u0444\u0438\u043a\u0430\u0442\u043e\u0440\u0430.}&amp; \\\\\n\n(5)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k \\in \\mathbb{Z})(b+c= ka) &amp;\\text{; \u0418\u0437 (4)}\\\\\n\n&amp;\\text{\u0417\u0430\u043c\u043a\u043d\u0443\u0442\u043e\u0441\u0442\u044c $\\mathbb{Z}$ \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u043e \u0441\u043b\u043e\u0436\u0435\u043d\u0438\u044f.}&amp; \\\\\n\n(6)&amp;\\{a\\mid b, a\\mid c\\}\\vdash a\\mid (b+c) &amp;\\text{; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438 (5)}\\\\\n\n(7)&amp;\\vdash (a\\mid b \\wedge a\\mid c) \\rightarrow a\\mid (b+c) &amp;\\text{; $\\wedge$-TD (6)}\\\\\n\n(8)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_1, k_2 \\in \\mathbb{Z})(b-c= (k_1-k_2)a) &amp;\\text{; \u0418\u0437 (3)}\\\\\n\n&amp;\\text{\u0410\u043b\u0433\u0435\u0431\u0440\u0430 \u0432\u043d\u0443\u0442\u0440\u0438 \u043a\u0432\u0430\u043d\u0442\u0438\u0444\u0438\u043a\u0430\u0442\u043e\u0440\u0430.}&amp; \\\\\n\n(9)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists \\overline{k} \\in \\mathbb{Z})(b-c= \\overline{k}a) &amp;\\text{; \u0418\u0437 (8)}\\\\\n\n&amp;\\text{\u0417\u0430\u043c\u043a\u043d\u0443\u0442\u043e\u0441\u0442\u044c $\\mathbb{Z}$ \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u043e \u0432\u044b\u0447\u0438\u0442\u0430\u043d\u0438\u044f.}&amp; \\\\\n\n(10)&amp;\\{a\\mid b, a\\mid c\\}\\vdash a\\mid (b-c) &amp;\\text{; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438 (9)}\\\\\n\n(11)&amp;\\vdash (a\\mid b \\wedge a\\mid c) \\rightarrow a\\mid (b-c) &amp;\\text{; $\\wedge$-TD (10)}\\\\\n\n(12)&amp;\\vdash (a\\mid b \\wedge a\\mid c) \\rightarrow \\left(a\\mid (b+c) \\wedge a\\mid (b-c)\\right) &amp;\\text{; $\\wedge$-\u0438\u043d\u0442\u0440. \u0432 \u043a\u043e\u043d\u0441\u0435\u043a\u0432\u0435\u043d\u0442\u0435 (7,11) }\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}<\/span>\n<\/li>\n<li><strong>\u0421\u043e\u0432\u043c\u0435\u0441\u0442\u0438\u043c\u043e\u0441\u0442\u044c \u0441 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f\u043c\u0438:<\/strong> \u0435\u0441\u043b\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, \u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (bc)<\/span><\/span> \u0434\u043b\u044f \u043b\u044e\u0431\u043e\u0433\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in\\mathbb{Z}<\/span><\/span>.<br \/>\n<u>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e<\/u>:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1)&amp; \\{a\\mid b\\}\\vdash (\\exists k\\in\\mathbb{Z})(b=ka) &amp;\\text{; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438, \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435}\\\\\n\n(2)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (\\exists k\\in\\mathbb{Z})(cb=cka) &amp;\\text{; \u0418\u0437 (1), $\\forall$-\u0438\u043d\u0442\u0440. (c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u044c\u043d\u043e)}\\\\\n\n&amp;\\text{\u0410\u043b\u0433\u0435\u0431\u0440\u0430 \u0432 }\\mathbb{Z}\\text{ \u0432\u043d\u0443\u0442\u0440\u0438 \u043a\u0432\u0430\u043d\u0442\u0438\u0444\u0438\u043a\u0430\u0442\u043e\u0440\u0430 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u043e\u0432\u0430\u043d\u0438\u044f.}&amp;\\\\\n\n(3)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (\\exists \\overline{k}\\in\\mathbb{Z})(cb=\\overline{k}a) &amp;\\text{; \u0418\u0437 (2), \u0437\u0430\u043c\u043a\u043d\u0443\u0442\u043e\u0441\u0442\u044c: }\\overline{k}=ck\\\\\n\n(4)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (a \\mid cb) &amp;\\text{; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438 (3)}\\\\\n\n(5)&amp; \\vdash a\\mid b \\rightarrow \\left(\\forall c \\in \\mathbb{Z}\\right) (a \\mid cb) &amp;\\text{; TD (4)}\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<\/ul>\n<h3><b>\u0422\u0435\u043e\u0440\u0435\u043c\u0430:<\/b> \u043e\u0446\u0435\u043d\u043a\u0430 \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u044f<\/h3>\n<p style=\"text-align: justify;\">\n\u0415\u0441\u043b\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\neq 0<\/span><\/span> \u0438 <span dir=\"ltr\"> <span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, \u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|\\le |b|<\/span><\/span>.\n<\/p>\n<p><b>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e:<\/b><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash b \\neq 0\n\n&amp; \\text{; \u041f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435} \\\\\n\n(2) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash (\\exists k \\in \\mathbb{Z}) (b=ka)\n\n&amp; \\text{; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438, \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435} \\\\\n\n(3) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash (\\exists k \\in \\mathbb{Z}) (|b|=|k||a|)\n\n&amp; \\text{; \u0421\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0430\u0431\u0441\u043e\u043b\u044e\u0442\u043d\u043e\u0439 \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u044b, \u0438\u0437 (2)} \\\\\n\n(4) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash (\\exists k \\in \\mathbb{Z}) (k\\neq 0 \\wedge |b|=|k||a|)\n\n&amp; \\text{; \u0418\u0437 (1,3)} \\\\\n\n(5) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash (\\exists k \\in \\mathbb{Z}) (1\\le |k| \\wedge |b|=|k||a|)\n\n&amp; \\text{; \u0418\u0437 (4), \u0435\u0441\u043b\u0438 }k\\neq 0\\Rightarrow |k|\\ge 1 \\\\\n\n(6) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash |a|\\le |b|\n\n&amp; \\text{; \u0418\u0437 (5)} \\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<p><a name=\"3\"><\/a><\/br><\/p>\n<h2>\u041f\u0440\u0435\u0434\u043b\u0430\u0433\u0430\u0435\u043c\u044b\u0435 \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f<\/h2>\n<ol>\n<li>\u041f\u043e\u043a\u0430\u0436\u0438\u0442\u0435, \u0447\u0442\u043e \u0442\u0435\u043e\u0440\u0435\u043c\u0430 \u00ab\u043e\u0446\u0435\u043d\u043a\u0430 \u0434\u0435\u043b\u0438\u0442\u0435\u043b\u044f\u00bb \u043d\u0435 \u043e\u0431\u044f\u0437\u0430\u0442\u0435\u043b\u044c\u043d\u043e \u0432\u0435\u0440\u043d\u0430, \u0435\u0441\u043b\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span>.<\/li>\n<li>\u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u0438 \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> \u043d\u0430 \u044d\u0442\u043e\u043c \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0435. \u0415\u0441\u043b\u0438 \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u044b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x,y\\in A<\/span><\/span> \u0442\u0430\u043a\u043e\u0432\u044b, \u0447\u0442\u043e <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u0441\u0432\u044f\u0437\u0430\u043d\u043e \u0441 <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> \u043f\u043e\u0441\u0440\u0435\u0434\u0441\u0442\u0432\u043e\u043c <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span>, \u0442\u043e \u044d\u0442\u043e \u0437\u0430\u043f\u0438\u0441\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u043a\u0430\u043a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\rho y<\/span><\/span>. \u041e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435 <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f <strong>\u0447\u0430\u0441\u0442\u0438\u0447\u043d\u044b\u043c \u043f\u043e\u0440\u044f\u0434\u043a\u043e\u043c<\/strong> \u043d\u0430 <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span>, \u0435\u0441\u043b\u0438:\n<p>a)<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x\\in A) (x\\rho x)<\/span><\/span>,<br \/>\nb) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x,y\\in A) ( (x\\rho y \\wedge y\\rho x) \\rightarrow x=y)<\/span><\/span><br \/>\nc) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x,y,z\\in A) ( (x\\rho y \\wedge y\\rho z) \\rightarrow x\\rho z)<\/span><\/span>.<\/p>\n<p>\u0414\u043e\u043a\u0430\u0436\u0438\u0442\u0435, \u0447\u0442\u043e \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435 \u0434\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u0438 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435\u043c \u0447\u0430\u0441\u0442\u0438\u0447\u043d\u043e\u0433\u043e \u043f\u043e\u0440\u044f\u0434\u043a\u0430 \u043d\u0430 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0435 \u0446\u0435\u043b\u044b\u0445 \u0447\u0438\u0441\u0435\u043b.<\/li>\n<li>\u0414\u043e\u043a\u0430\u0436\u0438\u0442\u0435 \u043f\u043e \u0438\u043d\u0434\u0443\u043a\u0446\u0438\u0438, \u0447\u0442\u043e \u0435\u0441\u043b\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b_1, a\\mid b_2, \\cdots, a\\mid b_n<\/span><\/span>, \u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid \\sum_{i=1}^n b_i x_i<\/span><\/span> \u0434\u043b\u044f \u043b\u044e\u0431\u043e\u0433\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{x_i\\}_{i=1}^n \\subset \\mathbb{Z}<\/span><\/span>. \u041a\u0440\u043e\u043c\u0435 \u0442\u043e\u0433\u043e, \u0434\u043e\u043a\u0430\u0436\u0438\u0442\u0435, \u0447\u0442\u043e \u0435\u0441\u043b\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b_i<\/span><\/span> \u043f\u0440\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i\\in \\{1,2,3,\\cdots, n\\}<\/span><\/span> \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> \u043c\u043e\u0436\u0435\u0442 \u0431\u044b\u0442\u044c \u0437\u0430\u043f\u0438\u0441\u0430\u043d\u043e \u043a\u0430\u043a \u043b\u0438\u043d\u0435\u0439\u043d\u0430\u044f \u043a\u043e\u043c\u0431\u0438\u043d\u0430\u0446\u0438\u044f \u044d\u0442\u0438\u0445 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b_i<\/span><\/span>, \u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>. <\/li>\n<li>\u0415\u0441\u043b\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>, \u043f\u043e\u043a\u0430\u0436\u0438\u0442\u0435, \u0447\u0442\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{x\\;:\\; d\\mid a\\}<\/span><\/span> \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043a\u043e\u043d\u0435\u0447\u043d\u044b\u043c.<\/li>\n<li>\u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0444\u0438\u043a\u0441\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u043e\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\in\\mathbb{Z}^+<\/span><\/span> \u0438 \u043f\u043e\u043b\u043e\u0436\u0438\u043c\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S=\\{d\\,:\\,d\\in\\mathbb{Z}^+ \\wedge d\\mid n\\}<\/span>\n<p>\u0414\u043e\u043a\u0430\u0436\u0438\u0442\u0435:<\/p>\n<ol>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d\\in S \\leftrightarrow n\/d\\in S<\/span><\/span><\/li>\n<li type=\"a\">\u0415\u0441\u043b\u0438 \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u044b <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> \u0443\u043f\u043e\u0440\u044f\u0434\u043e\u0447\u0435\u043d\u044b \u043f\u043e \u0432\u043e\u0437\u0440\u0430\u0441\u0442\u0430\u043d\u0438\u044e: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1=d_1 \\lt d_2 \\lt \\cdots \\lt d_t =n<\/span><\/span>, \u0442\u043e \u0441\u043e\u043e\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u044e\u0449\u0438\u0435 \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u044b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\mid d_i<\/span><\/span> \u043f\u0440\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i \\in \\{1,2,\\cdots, t\\}<\/span><\/span> \u0440\u0430\u0441\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u044b \u0432 \u043f\u043e\u0440\u044f\u0434\u043a\u0435 \u0443\u0431\u044b\u0432\u0430\u043d\u0438\u044f.<\/li>\n<\/ol>\n<\/li>\n<li>\u041f\u0443\u0441\u0442\u044c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}^+<\/span><\/span> \u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ab=c<\/span><\/span>. \u0414\u043e\u043a\u0430\u0436\u0438\u0442\u0435, \u0447\u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\min\\{a,b\\}\\le \\sqrt{c}<\/span><\/span>.<\/li>\n<li>\u0426\u0435\u043b\u043e\u0435 \u0447\u0438\u0441\u043b\u043e <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u043d\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u0447\u0435\u0442\u043d\u044b\u043c, \u0435\u0441\u043b\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2\\mid n<\/span><\/span>, \u0438 \u043d\u0435\u0447\u0435\u0442\u043d\u044b\u043c, \u0435\u0441\u043b\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2\\nmid n<\/span><\/span>. \u0414\u043e\u043a\u0430\u0436\u0438\u0442\u0435, \u0447\u0442\u043e \u0441\u0443\u043c\u043c\u0430 \u0438 \u0440\u0430\u0437\u043d\u043e\u0441\u0442\u044c:\n<ol>\n<li type=\"a\">\u0434\u0432\u0443\u0445 \u0447\u0435\u0442\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0447\u0435\u0442\u043d\u044b\u043c \u0447\u0438\u0441\u043b\u043e\u043c.<\/li>\n<li type=\"a\">\u0434\u0432\u0443\u0445 \u043d\u0435\u0447\u0435\u0442\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0447\u0435\u0442\u043d\u044b\u043c \u0447\u0438\u0441\u043b\u043e\u043c.<\/li>\n<li type=\"a\">\u0447\u0435\u0442\u043d\u043e\u0433\u043e \u0438 \u043d\u0435\u0447\u0435\u0442\u043d\u043e\u0433\u043e \u0447\u0438\u0441\u043b\u0430 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043d\u0435\u0447\u0435\u0442\u043d\u044b\u043c \u0447\u0438\u0441\u043b\u043e\u043c.<\/li>\n<\/ol>\n<\/li>\n<li>\u0415\u0441\u043b\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u2014 \u043d\u0435\u0447\u0435\u0442\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e, \u043e\u0442\u043b\u0438\u0447\u043d\u043e\u0435 \u043e\u0442 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\pm 1<\/span><\/span>, \u0434\u043e\u043a\u0430\u0436\u0438\u0442\u0435, \u0447\u0442\u043e <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u043d\u0435 \u043c\u043e\u0436\u0435\u0442 \u0434\u0435\u043b\u0438\u0442\u044c \u0434\u0432\u0430 \u043f\u043e\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0445 \u0447\u0435\u0442\u043d\u044b\u0445 \u0447\u0438\u0441\u043b\u0430.<\/li>\n<li>\u041f\u0443\u0441\u0442\u044c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b,n\\in\\mathbb{Z}<\/span><\/span> \u0442\u0430\u043a\u043e\u0432\u044b, \u0447\u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a-b|\\lt |n|<\/span><\/span>. \u0414\u043e\u043a\u0430\u0436\u0438\u0442\u0435, \u0447\u0442\u043e <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u043d\u0435 \u043c\u043e\u0436\u0435\u0442 \u0434\u0435\u043b\u0438\u0442\u044c \u043d\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span>, \u043d\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span>.<\/li>\n<li>\u041f\u0443\u0441\u0442\u044c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{Z}<\/span><\/span>. \u0414\u043e\u043a\u0430\u0436\u0438\u0442\u0435, \u0447\u0442\u043e:\n<ol>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall n \\in \\mathbb{Z})(a\\mid n) \\leftrightarrow a=\\pm 1<\/span><\/span><\/li>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall n \\in \\mathbb{Z})(n\\mid a) \\leftrightarrow a=0<\/span><\/span><\/li>\n<\/ol>\n<\/li>\n<li>\u041f\u0443\u0441\u0442\u044c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{Z}<\/span><\/span> \u0438 \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0438\u043c, \u0447\u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\neq 0<\/span><\/span>. \u041f\u043e\u043a\u0430\u0436\u0438\u0442\u0435, \u0447\u0442\u043e \u0438\u0437 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ac\\mid bc<\/span><\/span> \u0441\u043b\u0435\u0434\u0443\u0435\u0442, \u0447\u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>. <\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u0414\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u044c \u0414\u0435\u043b\u0438\u043c\u043e\u0441\u0442\u044c \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0440\u0435\u0430\u043b\u044c\u043d\u043e\u0439 \u043e\u0442\u043f\u0440\u0430\u0432\u043d\u043e\u0439 \u0442\u043e\u0447\u043a\u043e\u0439 \u0442\u0435\u043e\u0440\u0438\u0438 \u0447\u0438\u0441\u0435\u043b, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u043e\u043d\u0430 \u043f\u0440\u0435\u0432\u0440\u0430\u0449\u0430\u0435\u0442 \u0446\u0435\u043b\u044b\u0435 \u0447\u0438\u0441\u043b\u0430 \u0432 \u0441\u0438\u0441\u0442\u0435\u043c\u0443 \u0441\u043e \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u043e\u0439: \u0432\u044b \u043f\u0435\u0440\u0435\u0441\u0442\u0430\u0435\u0442\u0435 \u0440\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u0442\u044c \u0447\u0438\u0441\u043b\u0430 \u043a\u0430\u043a \u00ab\u043a\u043e\u043b\u0438\u0447\u0435\u0441\u0442\u0432\u0430\u00bb \u0438 \u043d\u0430\u0447\u0438\u043d\u0430\u0435\u0442\u0435 \u0432\u0438\u0434\u0435\u0442\u044c \u0438\u0445 \u043a\u0430\u043a \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u044b, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043b\u0438\u0431\u043e \u0441\u043e\u0433\u043b\u0430\u0441\u0443\u044e\u0442\u0441\u044f, \u043b\u0438\u0431\u043e \u043d\u0435 \u0441\u043e\u0433\u043b\u0430\u0441\u0443\u044e\u0442\u0441\u044f \u0434\u0440\u0443\u0433 \u0441 \u0434\u0440\u0443\u0433\u043e\u043c. \u0421 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u043e\u0434\u043d\u043e\u0439-\u0435\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0439 \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u044b, , \u043c\u043e\u0436\u043d\u043e \u0432\u044b\u0440\u0430\u0437\u0438\u0442\u044c \u043a\u0430\u043a \u043a\u0440\u0438\u0442\u0435\u0440\u0438\u0438 \u0443\u043f\u0440\u043e\u0449\u0435\u043d\u0438\u044f \u0438 \u0444\u0430\u043a\u0442\u043e\u0440\u0438\u0437\u0430\u0446\u0438\u0438, \u0442\u0430\u043a \u0438 \u0441\u0443\u0442\u044c \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u043f\u0440\u043e\u0446\u0435\u0434\u0443\u0440, 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