{"id":35669,"date":"2025-12-26T18:56:31","date_gmt":"2025-12-26T18:56:31","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35669"},"modified":"2025-12-26T18:56:31","modified_gmt":"2025-12-26T18:56:31","slug":"%d9%85%d8%a7-%d9%87%d9%8a-%d8%a7%d9%84%d9%82%d8%a7%d8%a8%d9%84%d9%8a%d8%a9-%d9%84%d9%84%d9%82%d8%b3%d9%85%d8%a9%d8%9f","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/ar\/%d9%85%d8%a7-%d9%87%d9%8a-%d8%a7%d9%84%d9%82%d8%a7%d8%a8%d9%84%d9%8a%d8%a9-%d9%84%d9%84%d9%82%d8%b3%d9%85%d8%a9%d8%9f\/","title":{"rendered":"\u0645\u0627 \u0647\u064a \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629\u061f"},"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>\u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629<\/h1>\n<p style=\"text-align:center;\"><em><br \/>\n\u062a\u064f\u0639\u064e\u062f\u0651 <strong>\u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629<\/strong> \u0646\u0642\u0637\u0629 \u0627\u0644\u0627\u0646\u0637\u0644\u0627\u0642 \u0627\u0644\u062d\u0642\u064a\u0642\u064a\u0629 \u0644\u0646\u0638\u0631\u064a\u0629 \u0627\u0644\u0623\u0639\u062f\u0627\u062f\u060c \u0644\u0623\u0646\u0647\u0627 \u062a\u062d\u0648\u0651\u0644 \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0635\u062d\u064a\u062d\u0629 \u0625\u0644\u0649 \u0646\u0638\u0627\u0645 \u0630\u064a \u0628\u0646\u064a\u0629: \u0625\u0630 \u0644\u0645 \u062a\u0639\u062f \u062a\u0646\u0638\u0631 \u0625\u0644\u0649 \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0628\u0648\u0635\u0641\u0647\u0627 \u00ab\u0643\u0645\u064a\u0627\u062a\u00bb \u0641\u062d\u0633\u0628\u060c \u0628\u0644 \u0628\u0627\u0639\u062a\u0628\u0627\u0631\u0647\u0627 \u0639\u0646\u0627\u0635\u0631 \u0625\u0645\u0651\u0627 \u0623\u0646 \u062a\u062a\u0648\u0627\u0641\u0642 \u0641\u064a\u0645\u0627 \u0628\u064a\u0646\u0647\u0627 \u0623\u0648 \u0644\u0627 \u062a\u062a\u0648\u0627\u0641\u0642. \u0641\u0645\u0646 \u062e\u0644\u0627\u0644 \u0628\u0646\u064a\u0629 \u0648\u0627\u062d\u062f\u0629 \u0641\u0642\u0637\u060c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>\u060c \u064a\u0645\u0643\u0646 \u0627\u0644\u062a\u0639\u0628\u064a\u0631 \u0639\u0646 \u0645\u0639\u0627\u064a\u064a\u0631 \u0627\u0644\u062a\u0628\u0633\u064a\u0637 \u0648\u0627\u0644\u062a\u062d\u0644\u064a\u0644 \u0625\u0644\u0649 \u0639\u0648\u0627\u0645\u0644\u060c \u0648\u0635\u0648\u0644\u064b\u0627 \u0625\u0644\u0649 \u062c\u0648\u0647\u0631 \u0628\u0639\u0636 \u0627\u0644\u0625\u062c\u0631\u0627\u0621\u0627\u062a \u0645\u062b\u0644 \u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0625\u0642\u0644\u064a\u062f\u0633\u060c \u0627\u0644\u062a\u064a \u062a\u064f\u0645\u0643\u0651\u0646 \u0645\u0646 \u062d\u0633\u0627\u0628 \u0627\u0644\u0642\u0648\u0627\u0633\u0645 \u0627\u0644\u0645\u0634\u062a\u0631\u0643\u0629 \u0627\u0644\u0639\u0638\u0645\u0649 \u0641\u064a \u062b\u0648\u0627\u0646\u064d \u062d\u062a\u0649 \u0645\u0639 \u0623\u0639\u062f\u0627\u062f \u0643\u0628\u064a\u0631\u0629. \u0641\u0636\u0644\u064b\u0627 \u0639\u0646 \u0630\u0644\u0643\u060c \u0641\u0647\u064a \u0627\u0644\u0623\u0633\u0627\u0633 \u0627\u0644\u062a\u0642\u0646\u064a \u0644\u0623\u0641\u0643\u0627\u0631 \u062a\u0638\u0647\u0631 \u0645\u0631\u0627\u0631\u064b\u0627 \u0641\u064a \u0627\u0644\u0631\u064a\u0627\u0636\u064a\u0627\u062a \u0627\u0644\u062a\u0637\u0628\u064a\u0642\u064a\u0629 \u0648\u0639\u0644\u0648\u0645 \u0627\u0644\u062d\u0627\u0633\u0648\u0628: \u0627\u0644\u062a\u5408\u540c\u0627\u062a\u060c \u0648\u0627\u0644\u062d\u0633\u0627\u0628\u064a\u0627\u062a \u0627\u0644\u0645\u0639\u064a\u0627\u0631\u064a\u0629\u060c \u0648\u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u062a\u062d\u0642\u0642\u060c \u0648\u0627\u0644\u0634\u0641\u0640\u0631\u0627\u062a\u060c \u0648(\u0644\u0627\u062d\u0642\u064b\u0627) \u0639\u0644\u0645 \u0627\u0644\u062a\u0634\u0641\u064a\u0631. \u0625\u0646 \u0625\u062a\u0642\u0627\u0646 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 \u064a\u0639\u0646\u064a\u060c \u0641\u064a \u062c\u0648\u0647\u0631\u0647\u060c \u062a\u0639\u0644\u0651\u0645 \u0627\u0643\u062a\u0634\u0627\u0641 \u0623\u0646\u0645\u0627\u0637 \u063a\u064a\u0631 \u0645\u0631\u0626\u064a\u0629 \u0641\u064a \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0635\u062d\u064a\u062d\u0629 \u0648\u062a\u062d\u0648\u064a\u0644\u0647\u0627 \u0625\u0644\u0649 \u0625\u062c\u0631\u0627\u0621\u0627\u062a \u062a\u0639\u0645\u0644 \u062f\u0627\u0626\u0645\u064b\u0627.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><b>\u0623\u0647\u062f\u0627\u0641 \u0627\u0644\u062a\u0639\u0644\u0651\u0645<\/b><br \/>\n\u0639\u0646\u062f \u0627\u0644\u0627\u0646\u062a\u0647\u0627\u0621 \u0645\u0646 \u0647\u0630\u0627 \u0627\u0644\u0645\u0644\u062e\u0635 \u0633\u064a\u0643\u0648\u0646 \u0627\u0644\u0637\u0627\u0644\u0628 \u0642\u0627\u062f\u0631\u064b\u0627 \u0639\u0644\u0649:\n<\/p>\n<ol>\n<li><strong>\u0641\u0647\u0645<\/strong> \u0639\u0644\u0627\u0642\u0629 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 \u0628\u064a\u0646 \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0635\u062d\u064a\u062d\u0629.<\/li>\n<li><strong>\u0641\u0647\u0645<\/strong> \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 \u0648\u062e\u0635\u0627\u0626\u0635\u0647\u0627.<\/li>\n<li><strong>\u062a\u0637\u0648\u064a\u0631<\/strong> \u0628\u0631\u0627\u0647\u064a\u0646 \u0631\u064a\u0627\u0636\u064a\u0629 \u0644\u0646\u062a\u0627\u0626\u062c \u0648\u0646\u0638\u0631\u064a\u0627\u062a \u0645\u062a\u0639\u0644\u0642\u0629 \u0628\u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><b><u>\u0641\u0647\u0631\u0633 \u0627\u0644\u0645\u062d\u062a\u0648\u064a\u0627\u062a<\/u><\/b><br \/>\n<a href=\"#1\">\u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629<\/a><br \/>\n<a href=\"#2\">\u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u0644\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629<\/a><br \/>\n<a href=\"#3\">\u062a\u0645\u0627\u0631\u064a\u0646 \u0645\u0642\u062a\u0631\u062d\u0629<\/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>\u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629<\/h2>\n<p style=\"text-align: justify;\">\n\u062a\u064f\u0635\u0628\u062d \u0627\u0644\u0641\u0643\u0631\u0629 \u063a\u064a\u0631 \u0627\u0644\u0631\u0633\u0645\u064a\u0629 \u00ab<em>a \u064a\u0642\u0633\u0645 b<\/em>\u00bb \u062f\u0642\u064a\u0642\u0629 \u0639\u0646\u062f\u0645\u0627 \u0646\u0639\u0628\u0651\u0631 \u0639\u0646\u0647\u0627 \u0628\u0648\u0635\u0641\u0647\u0627 \u0639\u0644\u0627\u0642\u0629 \u0628\u064a\u0646 \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0635\u062d\u064a\u062d\u0629. \u0646\u0642\u0648\u0644 \u0625\u0646 \u0639\u062f\u062f\u064b\u0627 \u0635\u062d\u064a\u062d\u064b\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> <strong>\u064a\u0642\u0633\u0645<\/strong> \u0639\u062f\u062f\u064b\u0627 \u0635\u062d\u064a\u062d\u064b\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u0625\u0630\u0627 \u0623\u0645\u0643\u0646 \u0643\u062a\u0627\u0628\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u0639\u0644\u0649 \u0623\u0646\u0647 \u0645\u0636\u0627\u0639\u0641 \u062f\u0642\u064a\u0642 \u0644\u0640 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>. \u064a\u0634\u0643\u0651\u0644 \u0647\u0630\u0627 \u0627\u0644\u062a\u0639\u0631\u064a\u0641 \u0623\u0633\u0627\u0633 \u0628\u0642\u064a\u0629 \u0647\u0630\u0627 \u0627\u0644\u0645\u0644\u062e\u0635\u060c \u0644\u0623\u0646\u0647 \u064a\u062d\u0648\u0651\u0644 \u0639\u0628\u0627\u0631\u0627\u062a \u0645\u0646 \u0642\u0628\u064a\u0644 \u00ab\u064a\u0646\u0627\u0633\u0628 \u062a\u0645\u0627\u0645\u064b\u0627\u00bb \u0625\u0644\u0649 \u0645\u0639\u064a\u0627\u0631 \u0642\u0627\u0628\u0644 \u0644\u0644\u062a\u062d\u0642\u0642.\n<\/p>\n<p style=\"text-align: justify;\">\n<strong>\u062a\u0639\u0631\u064a\u0641.<\/strong> \u0644\u062a\u0643\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}<\/span><\/span> \u0645\u0639 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>. \u0646\u0642\u0648\u0644 \u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u064a\u0642\u0633\u0645 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span>\u060c \u0648\u0646\u0643\u062a\u0628 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>\u060c \u0625\u0630\u0627 \u0648\u0641\u0642\u0637 \u0625\u0630\u0627 \u0648\u064f\u062c\u062f \u0639\u062f\u062f \u0635\u062d\u064a\u062d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> \u0628\u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=ka<\/span><\/span>. \u0648\u0641\u064a \u063a\u064a\u0631 \u0630\u0644\u0643 \u0646\u0643\u062a\u0628 <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\u0641\u064a \u0647\u0630\u0627 \u0627\u0644\u062a\u0639\u0631\u064a\u0641\u060c \u064a\u064f\u0633\u0645\u0651\u0649 \u0627\u0644\u0639\u062f\u062f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> <strong>\u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629<\/strong> (\u0623\u0648 \u0627\u0644\u0639\u0627\u0645\u0644) \u0627\u0644\u0645\u0631\u062a\u0628\u0637 \u0628\u0639\u0644\u0627\u0642\u0629 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629. \u0641\u0639\u0644\u0649 \u0633\u0628\u064a\u0644 \u0627\u0644\u0645\u062b\u0627\u0644\u060c \u0625\u0646 \u0627\u0644\u0642\u0648\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">6\\mid 42<\/span><\/span> \u064a\u0643\u0627\u0641\u0626 \u0627\u0644\u0642\u0648\u0644 \u0628\u0648\u062c\u0648\u062f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> \u0628\u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">42=6k<\/span><\/span>\u061b \u0648\u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u062d\u0627\u0644\u0629 \u064a\u0643\u0641\u064a \u0623\u062e\u0630 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=7<\/span><\/span>.\n<\/p>\n<h3>\u0645\u0646 \u0627\u0644\u0645\u0647\u0645 \u0645\u0631\u0627\u0639\u0627\u0629<\/h3>\n<ul>\n<li>\n    \u0625\u0646 \u0627\u0644\u0634\u0631\u0637 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span> \u0634\u0631\u0637\u064c \u0623\u0633\u0627\u0633\u064a\u060c \u0644\u0623\u0646\u0647 \u0644\u0648 \u062d\u0627\u0648\u0644\u0646\u0627 \u0627\u0644\u0633\u0645\u0627\u062d \u0628\u0640 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=0<\/span><\/span>\u060c \u0641\u0625\u0646 \u0634\u0631\u0637 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 \u0633\u064a\u0641\u0631\u0636 \u0648\u062c\u0648\u062f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> \u0628\u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0\\cdot k<\/span><\/span>. \u063a\u064a\u0631 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\cdot k=0<\/span><\/span> \u0644\u0643\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span>\u060c \u0648\u0645\u0646 \u062b\u0645 \u062a\u0643\u0648\u0646 \u0627\u0644\u0625\u0645\u0643\u0627\u0646\u064a\u0629 \u0627\u0644\u0648\u062d\u064a\u062f\u0629 \u0647\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span>. \u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u062d\u0627\u0644\u0629 \u0644\u0627 \u064a\u0648\u062c\u062f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> \u00ab\u0645\u062d\u062f\u062f\u00bb \u0628\u0627\u0644\u0639\u0644\u0627\u0642\u0629\u060c \u0625\u0630 \u0625\u0646 \u0623\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> \u064a\u062d\u0642\u0642 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=0\\cdot k<\/span><\/span>. \u0648\u0628\u0639\u0628\u0627\u0631\u0629 \u0623\u062e\u0631\u0649\u060c \u0641\u0625\u0646 \u0627\u0644\u062a\u0639\u0628\u064a\u0631 \u063a\u064a\u0631 \u0627\u0644\u0631\u0633\u0645\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=b\/a<\/span><\/span> \u064a\u062a\u062d\u0648\u0644 \u0625\u0644\u0649 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0\/0<\/span><\/span>\u060c \u0648\u0647\u0648 \u063a\u064a\u0631 \u0645\u0639\u0631\u0651\u0641. \u0648\u0644\u062a\u0641\u0627\u062f\u064a \u0647\u0630\u0627 \u0627\u0644\u0627\u0646\u062d\u0644\u0627\u0644 (\u062d\u064a\u062b \u062a\u0641\u0642\u062f \u0641\u0643\u0631\u0629 \u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629 \u0645\u0639\u0646\u0627\u0647\u0627)\u060c \u064a\u064f\u0634\u062a\u0631\u0637 \u0623\u0646 \u064a\u0643\u0648\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>. \u0648\u0644\u0647\u0630\u0627 \u0627\u0644\u0633\u0628\u0628 \u0644\u0627 \u062a\u064f\u0639\u062f \u0627\u0644\u0639\u0644\u0627\u0642\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\mid b<\/span><\/span> \u0635\u0627\u0644\u062d\u0629.\n<\/li>\n<li>\n        \u0641\u064a \u0627\u0644\u0645\u0642\u0627\u0628\u0644\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid 0<\/span><\/span> \u0635\u062d\u064a\u062d\u0629 \u0644\u0643\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{Z}<\/span><\/span> \u0645\u0639 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>\u060c \u0625\u0630 \u064a\u0643\u0641\u064a \u0623\u062e\u0630 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span> \u0641\u0646\u062d\u0635\u0644 \u0639\u0644\u0649 <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\u0648\u064a\u0646\u062a\u062c \u0639\u0646 \u0647\u0630\u0627 \u0627\u0644\u062a\u0639\u0631\u064a\u0641 \u062a\u0643\u0627\u0641\u0624 \u0633\u0646\u0633\u062a\u062e\u062f\u0645\u0647 \u0628\u0635\u0648\u0631\u0629 \u0645\u062a\u0643\u0631\u0631\u0629: \u0627\u0644\u0642\u0648\u0644 \u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> \u064a\u0643\u0627\u0641\u0626 \u0627\u0644\u0642\u0648\u0644 \u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u064a\u0646\u062a\u0645\u064a \u0625\u0644\u0649 \u0645\u062c\u0645\u0648\u0639\u0629 \u0627\u0644\u0645\u0636\u0627\u0639\u0641\u0627\u062a \u0627\u0644\u0635\u062d\u064a\u062d\u0629 \u0644\u0640 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>\u060c \u0623\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\in a\\mathbb{Z}<\/span><\/span>\u060c \u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mathbb{Z}=\\{ak:\\,k\\in\\mathbb{Z}\\}<\/span><\/span>. \u0648\u062a\u064f\u0628\u0631\u0632 \u0647\u0630\u0647 \u0627\u0644\u0635\u064a\u0627\u063a\u0629 \u0623\u0646 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 \u0644\u064a\u0633\u062a \u00ab\u062d\u064a\u0644\u0629\u00bb\u060c \u0628\u0644 \u0637\u0631\u064a\u0642\u0629 \u0644\u0648\u0635\u0641 \u0645\u062c\u0645\u0648\u0639\u0627\u062a \u062c\u0632\u0626\u064a\u0629 \u0639\u0627\u0644\u064a\u0629 \u0627\u0644\u062a\u0646\u0638\u064a\u0645 \u062f\u0627\u062e\u0644 <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>\u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u0644\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629<\/h2>\n<ul>\n<li><strong>\u0627\u0644\u0627\u0646\u0639\u0643\u0627\u0633\u064a\u0629:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid a<\/span><\/span>.<br \/>\n<u>\u0627\u0644\u0628\u0631\u0647\u0627\u0646<\/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{; \u0627\u0644\u0639\u0646\u0635\u0631 \u0627\u0644\u0645\u062d\u0627\u064a\u062f \u0644\u0644\u0636\u0631\u0628 \u0641\u064a $\\mathbb{Z}$}\\\\\n\n(2)&amp;\\vdash(\\exists k \\in \\mathbb{Z})(a=ka) &amp;\\text{; \u0625\u062f\u062e\u0627\u0644 \u0627\u0644\u0648\u062c\u0648\u062f\u064a (1)}\\\\\n\n(3) &amp;\\vdash a \\mid a &amp;\\text{; \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 (2)} \\\\\n\n&amp;\\blacksquare &amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<li><strong>\u0627\u0644\u062a\u0639\u062f\u0651\u064a:<\/strong> \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\mid c<\/span><\/span>\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>.\n<p><u>\u0627\u0644\u0628\u0631\u0647\u0627\u0646<\/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{; \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629\u060c \u0627\u0641\u062a\u0631\u0627\u0636}\\\\\n\n(2)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_2\\in\\mathbb{Z})(c=k_2b)  &amp;\\text{; \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629\u060c \u0627\u0641\u062a\u0631\u0627\u0636}\\\\\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{; \u062a\u062c\u0645\u064a\u0639 \u0627\u0644\u0648\u062c\u0648\u062f\u064a (1\u060c2)}\\\\\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{; \u0645\u0646 (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{; \u0645\u0646 (4)}\\\\\n\n&amp;\\text{\u062c\u0628\u0631 \u062f\u0627\u062e\u0644 \u0627\u0644\u0645\u064f\u0643\u0645\u0651\u0650\u0645}&amp; \\\\\n\n(6)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k\\in\\mathbb{Z})( c=ka)  &amp;\\text{; \u0645\u0646 (5)}\\\\\n\n&amp;\\text{\u0627\u0646\u063a\u0644\u0627\u0642 $\\mathbb{Z}$ \u062a\u062d\u062a \u0627\u0644\u0636\u0631\u0628}&amp; \\\\\n\n(7)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash a\\mid c  &amp;\\text{; \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 (6)}\\\\\n\n(8)&amp; \\vdash (a\\mid b \\wedge  b\\mid c) \\rightarrow  a\\mid c  &amp;\\text{; \u0646\u0642\u0644 \u0627\u0644\u0639\u0637\u0641 \u0625\u0644\u0649 \u0627\u0644\u062f\u0644\u0627\u0644\u0629 (7)}\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<li><strong>\u0627\u0644\u062a\u0648\u0627\u0641\u0642 \u0645\u0639 \u0627\u0644\u062c\u0645\u0639 \u0648\u0627\u0644\u0637\u0631\u062d:<\/strong> \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (b+c)<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (b-c)<\/span><\/span>.<br \/>\n<u>\u0627\u0644\u0628\u0631\u0647\u0627\u0646<\/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{; \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629\u060c \u0627\u0641\u062a\u0631\u0627\u0636}\\\\\n\n(2)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_2 \\in \\mathbb{Z})(c=k_2 a) &amp;\\text{; \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629\u060c \u0627\u0641\u062a\u0631\u0627\u0636}\\\\\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{; \u062a\u062c\u0645\u064a\u0639 \u0627\u0644\u0648\u062c\u0648\u062f\u064a (1\u060c2)}\\\\\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{; \u0645\u0646 (3)}\\\\\n\n&amp;\\text{\u062c\u0628\u0631 \u062f\u0627\u062e\u0644 \u0627\u0644\u0645\u064f\u0643\u0645\u0651\u0650\u0645.}&amp; \\\\\n\n(5)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k \\in \\mathbb{Z})(b+c= ka) &amp;\\text{; \u0645\u0646 (4)}\\\\\n\n&amp;\\text{\u0627\u0646\u063a\u0644\u0627\u0642 $\\mathbb{Z}$ \u062a\u062d\u062a \u0627\u0644\u062c\u0645\u0639.}&amp; \\\\\n\n(6)&amp;\\{a\\mid b, a\\mid c\\}\\vdash a\\mid (b+c) &amp;\\text{; \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 (5)}\\\\\n\n(7)&amp;\\vdash (a\\mid b \\wedge a\\mid c) \\rightarrow a\\mid (b+c) &amp;\\text{; \u0646\u0642\u0644 \u0627\u0644\u0639\u0637\u0641 \u0625\u0644\u0649 \u0627\u0644\u062f\u0644\u0627\u0644\u0629 (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{; \u0645\u0646 (3)}\\\\\n\n&amp;\\text{\u062c\u0628\u0631 \u062f\u0627\u062e\u0644 \u0627\u0644\u0645\u064f\u0643\u0645\u0651\u0650\u0645.}&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{; \u0645\u0646 (8)}\\\\\n\n&amp;\\text{\u0627\u0646\u063a\u0644\u0627\u0642 $\\mathbb{Z}$ \u062a\u062d\u062a \u0627\u0644\u0637\u0631\u062d.}&amp; \\\\\n\n(10)&amp;\\{a\\mid b, a\\mid c\\}\\vdash a\\mid (b-c) &amp;\\text{; \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 (9)}\\\\\n\n(11)&amp;\\vdash (a\\mid b \\wedge a\\mid c) \\rightarrow a\\mid (b-c) &amp;\\text{; \u0646\u0642\u0644 \u0627\u0644\u0639\u0637\u0641 \u0625\u0644\u0649 \u0627\u0644\u062f\u0644\u0627\u0644\u0629 (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{; \u0625\u062f\u062e\u0627\u0644 \u0627\u0644\u0639\u0637\u0641 \u0641\u064a \u0627\u0644\u0646\u062a\u064a\u062c\u0629 (7\u060c11)}\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}<\/span>\n<\/li>\n<li><strong>\u0627\u0644\u062a\u0648\u0627\u0641\u0642 \u0645\u0639 \u0627\u0644\u0636\u0631\u0628:<\/strong> \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (bc)<\/span><\/span> \u0644\u0643\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in\\mathbb{Z}<\/span><\/span>.<br \/>\n<u>\u0627\u0644\u0628\u0631\u0647\u0627\u0646<\/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{; \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629\u060c \u0627\u0641\u062a\u0631\u0627\u0636}\\\\\n\n(2)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (\\exists k\\in\\mathbb{Z})(cb=cka) &amp;\\text{; \u0645\u0646 (1)\u060c \u0625\u062f\u062e\u0627\u0644 \u0627\u0644\u0643\u0644\u064a (c \u0627\u0639\u062a\u0628\u0627\u0637\u064a)}\\\\\n\n&amp;\\text{\u062c\u0628\u0631 \u0641\u064a }\\mathbb{Z}\\text{ \u062f\u0627\u062e\u0644 \u0627\u0644\u0645\u064f\u0643\u0645\u0651\u0650\u0645 \u0627\u0644\u0648\u062c\u0648\u062f\u064a.}&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{; \u0645\u0646 (2)\u060c \u0627\u0644\u0627\u0646\u063a\u0644\u0627\u0642: }\\overline{k}=ck\\\\\n\n(4)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (a \\mid cb) &amp;\\text{; \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 (3)}\\\\\n\n(5)&amp; \\vdash a\\mid b \\rightarrow \\left(\\forall c \\in \\mathbb{Z}\\right) (a \\mid cb) &amp;\\text{; \u0646\u0642\u0644 \u0627\u0644\u062f\u0644\u0627\u0644\u0629 (4)}\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<\/ul>\n<h3><b>\u0646\u0638\u0631\u064a\u0629:<\/b> \u062d\u062f\u0651 \u0627\u0644\u0642\u0627\u0633\u0645<\/h3>\n<p style=\"text-align: justify;\">\n\u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\neq 0<\/span><\/span> \u0648 <span dir=\"ltr\"> <span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|\\le |b|<\/span><\/span>.\n<\/p>\n<p><b>\u0627\u0644\u0628\u0631\u0647\u0627\u0646:<\/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{; \u0627\u0641\u062a\u0631\u0627\u0636} \\\\\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{; \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629\u060c \u0627\u0641\u062a\u0631\u0627\u0636} \\\\\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{; \u062e\u0627\u0635\u064a\u0629 \u0627\u0644\u0642\u064a\u0645\u0629 \u0627\u0644\u0645\u0637\u0644\u0642\u0629\u060c \u0645\u0646 (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{; \u0645\u0646 (1\u060c3)} \\\\\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{; \u0645\u0646 (4)\u060c \u0625\u0630 }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{; \u0645\u0646 (5)} \\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<p><a name=\"3\"><\/a><\/br><\/p>\n<h2>\u062a\u0645\u0627\u0631\u064a\u0646 \u0645\u0642\u062a\u0631\u062d\u0629<\/h2>\n<ol>\n<li>\u0628\u064a\u0651\u0646 \u0623\u0646 \u0646\u0638\u0631\u064a\u0629 \u00ab\u062d\u062f\u0651 \u0627\u0644\u0642\u0627\u0633\u0645\u00bb \u0644\u064a\u0633\u062a \u0635\u062d\u064a\u062d\u0629 \u0628\u0627\u0644\u0636\u0631\u0648\u0631\u0629 \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span>.<\/li>\n<li>\u0644\u0646\u0639\u062a\u0628\u0631 \u0645\u062c\u0645\u0648\u0639\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u0648\u0639\u0644\u0627\u0642\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> \u0639\u0644\u0649 \u0647\u0630\u0647 \u0627\u0644\u0645\u062c\u0645\u0648\u0639\u0629. \u0625\u0630\u0627 \u0643\u0627\u0646 \u0627\u0644\u0639\u0646\u0635\u0631\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x,y\\in A<\/span><\/span> \u0628\u062d\u064a\u062b \u0625\u0646 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u0645\u0631\u062a\u0628\u0637 \u0628\u0640 <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> \u0628\u0648\u0627\u0633\u0637\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span>\u060c \u0641\u0625\u0646\u0646\u0627 \u0646\u0643\u062a\u0628 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\rho y<\/span><\/span><\/span>. \u0648\u062a\u064f\u0642\u0627\u0644 \u0627\u0644\u0639\u0644\u0627\u0642\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> \u0625\u0646\u0647\u0627 <strong>\u0639\u0644\u0627\u0642\u0629 \u062a\u0631\u062a\u064a\u0628 \u062c\u0632\u0626\u064a<\/strong> \u0639\u0644\u0649 <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u0625\u0630\u0627 \u062a\u062d\u0642\u0642\u062a \u0627\u0644\u0634\u0631\u0648\u0637 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:\n<p>a)<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x\\in A) (x\\rho x)<\/span><\/span>\u060c<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>\u0623\u062b\u0628\u062a \u0623\u0646 \u0639\u0644\u0627\u0642\u0629 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 \u0647\u064a \u0639\u0644\u0627\u0642\u0629 \u062a\u0631\u062a\u064a\u0628 \u062c\u0632\u0626\u064a \u0639\u0644\u0649 \u0645\u062c\u0645\u0648\u0639\u0629 \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0635\u062d\u064a\u062d\u0629.<\/li>\n<li>\u0623\u062b\u0628\u062a \u0628\u0627\u0644\u0627\u0633\u062a\u0642\u0631\u0627\u0621 \u0623\u0646\u0647 \u0625\u0630\u0627 \u0643\u0627\u0646 <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>\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid \\sum_{i=1}^n b_i x_i<\/span><\/span> \u0644\u0623\u064a \u0645\u062c\u0645\u0648\u0639\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{x_i\\}_{i=1}^n \\subset \\mathbb{Z}<\/span><\/span>. \u0643\u0645\u0627 \u0623\u062b\u0628\u062a \u0623\u0646\u0647 \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b_i<\/span><\/span>\u060c \u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i\\in \\{1,2,3,\\cdots, n\\}<\/span><\/span> \u0648\u0643\u0627\u0646 <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> \u064a\u0645\u0643\u0646 \u0643\u062a\u0627\u0628\u062a\u0647 \u0639\u0644\u0649 \u0634\u0643\u0644 \u062a\u0631\u0643\u064a\u0628 \u062e\u0637\u064a \u0644\u0647\u0630\u0647 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b_i<\/span><\/span>\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>. <\/li>\n<li>\u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>\u060c \u0628\u064a\u0651\u0646 \u0623\u0646 \u0627\u0644\u0645\u062c\u0645\u0648\u0639\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{x\\;:\\; d\\mid a\\}<\/span><\/span> \u0645\u062c\u0645\u0648\u0639\u0629 \u0645\u0646\u062a\u0647\u064a\u0629.<\/li>\n<li>\u0644\u062a\u0643\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\in\\mathbb{Z}^+<\/span><\/span> \u0639\u062f\u062f\u064b\u0627 \u062b\u0627\u0628\u062a\u064b\u0627\u060c \u0648\u0644\u062a\u0643\u0646\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>\u0623\u062b\u0628\u062a \u0645\u0627 \u064a\u0644\u064a:<\/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\">\u0625\u0630\u0627 \u0631\u064f\u062a\u0651\u0628\u062a \u0639\u0646\u0627\u0635\u0631 <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> \u062a\u0631\u062a\u064a\u0628\u064b\u0627 \u062a\u0635\u0627\u0639\u062f\u064a\u064b\u0627: <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>\u060c \u0641\u0625\u0646 \u0627\u0644\u0639\u0646\u0627\u0635\u0631 \u0627\u0644\u0645\u0642\u0627\u0628\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\mid d_i<\/span><\/span> \u0645\u0639 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i \\in \\{1,2,\\cdots, t\\}<\/span><\/span> \u062a\u0643\u0648\u0646 \u0645\u0631\u062a\u0628\u0629 \u062a\u0631\u062a\u064a\u0628\u064b\u0627 \u062a\u0646\u0627\u0632\u0644\u064a\u064b\u0627.<\/li>\n<\/ol>\n<\/li>\n<li>\u0627\u0641\u062a\u0631\u0636 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}^+<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ab=c<\/span><\/span>. \u0623\u062b\u0628\u062a \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\min\\{a,b\\}\\le \\sqrt{c}<\/span><\/span>.<\/li>\n<li>\u064a\u064f\u0642\u0627\u0644 \u0639\u0646 \u0639\u062f\u062f \u0635\u062d\u064a\u062d <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u0625\u0646\u0647 \u0632\u0648\u062c\u064a \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2\\mid n<\/span><\/span>\u060c \u0648\u0641\u0631\u062f\u064a \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2\\nmid n<\/span><\/span>. \u0623\u062b\u0628\u062a \u0623\u0646 \u0645\u062c\u0645\u0648\u0639 \u0648\u0641\u0631\u0642:\n<ol>\n<li type=\"a\">\u0639\u062f\u062f\u064a\u0646 \u0632\u0648\u062c\u064a\u064a\u0646 \u0647\u0648 \u0639\u062f\u062f \u0632\u0648\u062c\u064a.<\/li>\n<li type=\"a\">\u0639\u062f\u062f\u064a\u0646 \u0641\u0631\u062f\u064a\u064a\u0646 \u0647\u0648 \u0639\u062f\u062f \u0632\u0648\u062c\u064a.<\/li>\n<li type=\"a\">\u0639\u062f\u062f \u0632\u0648\u062c\u064a \u0648\u0639\u062f\u062f \u0641\u0631\u062f\u064a \u0647\u0648 \u0639\u062f\u062f \u0641\u0631\u062f\u064a.<\/li>\n<\/ol>\n<\/li>\n<li>\u0625\u0630\u0627 \u0643\u0627\u0646 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u0639\u062f\u062f\u064b\u0627 \u0641\u0631\u062f\u064a\u064b\u0627 \u0645\u062e\u062a\u0644\u0641\u064b\u0627 \u0639\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\pm 1<\/span><\/span>\u060c \u0641\u0623\u062b\u0628\u062a \u0623\u0646 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u0644\u0627 \u064a\u0645\u0643\u0646\u0647 \u0623\u0646 \u064a\u0642\u0633\u0645 \u0639\u062f\u062f\u064a\u0646 \u0632\u0648\u062c\u064a\u064a\u0646 \u0645\u062a\u062a\u0627\u0644\u064a\u064a\u0646.<\/li>\n<li>\u0644\u062a\u0643\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b,n\\in\\mathbb{Z}<\/span><\/span> \u0628\u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a-b|\\lt |n|<\/span><\/span>. \u0623\u062b\u0628\u062a \u0623\u0646 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u0644\u0627 \u064a\u0645\u0643\u0646\u0647 \u0623\u0646 \u064a\u0642\u0633\u0645 \u0644\u0627 <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> \u0648\u0644\u0627 <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span>.<\/li>\n<li>\u0627\u0641\u062a\u0631\u0636 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{Z}<\/span><\/span>. \u0623\u062b\u0628\u062a \u0645\u0627 \u064a\u0644\u064a:\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>\u0644\u062a\u0643\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{Z}<\/span><\/span> \u0648\u0644\u0646\u0641\u0631\u0636 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\neq 0<\/span><\/span>. \u0628\u064a\u0651\u0646 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ac\\mid bc<\/span><\/span> \u064a\u0642\u062a\u0636\u064a \u0623\u0646 <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>\u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 \u062a\u064f\u0639\u064e\u062f\u0651 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 \u0646\u0642\u0637\u0629 \u0627\u0644\u0627\u0646\u0637\u0644\u0627\u0642 \u0627\u0644\u062d\u0642\u064a\u0642\u064a\u0629 \u0644\u0646\u0638\u0631\u064a\u0629 \u0627\u0644\u0623\u0639\u062f\u0627\u062f\u060c \u0644\u0623\u0646\u0647\u0627 \u062a\u062d\u0648\u0651\u0644 \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0635\u062d\u064a\u062d\u0629 \u0625\u0644\u0649 \u0646\u0638\u0627\u0645 \u0630\u064a \u0628\u0646\u064a\u0629: \u0625\u0630 \u0644\u0645 \u062a\u0639\u062f \u062a\u0646\u0638\u0631 \u0625\u0644\u0649 \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0628\u0648\u0635\u0641\u0647\u0627 \u00ab\u0643\u0645\u064a\u0627\u062a\u00bb \u0641\u062d\u0633\u0628\u060c \u0628\u0644 \u0628\u0627\u0639\u062a\u0628\u0627\u0631\u0647\u0627 \u0639\u0646\u0627\u0635\u0631 \u0625\u0645\u0651\u0627 \u0623\u0646 \u062a\u062a\u0648\u0627\u0641\u0642 \u0641\u064a\u0645\u0627 \u0628\u064a\u0646\u0647\u0627 \u0623\u0648 \u0644\u0627 \u062a\u062a\u0648\u0627\u0641\u0642. \u0641\u0645\u0646 \u062e\u0644\u0627\u0644 \u0628\u0646\u064a\u0629 \u0648\u0627\u062d\u062f\u0629 \u0641\u0642\u0637\u060c \u060c \u064a\u0645\u0643\u0646 \u0627\u0644\u062a\u0639\u0628\u064a\u0631 \u0639\u0646 \u0645\u0639\u0627\u064a\u064a\u0631 \u0627\u0644\u062a\u0628\u0633\u064a\u0637 \u0648\u0627\u0644\u062a\u062d\u0644\u064a\u0644 \u0625\u0644\u0649 \u0639\u0648\u0627\u0645\u0644\u060c \u0648\u0635\u0648\u0644\u064b\u0627 \u0625\u0644\u0649 \u062c\u0648\u0647\u0631 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":35642,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":19,"footnotes":""},"categories":[565,1408],"tags":[],"class_list":["post-35669","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-565","category-1408"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>\u0645\u0627 \u0647\u064a \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629\u061f - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"\u0645\u0627 \u0647\u064a \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 \u0641\u064a \u0646\u0638\u0631\u064a\u0629 \u0627\u0644\u0623\u0639\u062f\u0627\u062f\u061f \u062a\u0639\u0631\u064a\u0641 \u0631\u064a\u0627\u0636\u064a\u060c 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