{"id":35678,"date":"2025-12-26T18:56:30","date_gmt":"2025-12-26T18:56:30","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35678"},"modified":"2025-12-26T18:56:30","modified_gmt":"2025-12-26T18:56:30","slug":"%e0%a4%95%e0%a5%8d%e0%a4%af%e0%a4%be-%e0%a4%b9%e0%a5%88-%e0%a4%b5%e0%a4%bf%e0%a4%ad%e0%a4%be%e0%a4%9c%e0%a5%8d%e0%a4%af%e0%a4%a4%e0%a4%be","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/hi\/%e0%a4%95%e0%a5%8d%e0%a4%af%e0%a4%be-%e0%a4%b9%e0%a5%88-%e0%a4%b5%e0%a4%bf%e0%a4%ad%e0%a4%be%e0%a4%9c%e0%a5%8d%e0%a4%af%e0%a4%a4%e0%a4%be\/","title":{"rendered":"\u0915\u094d\u092f\u093e \u0939\u0948 \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e?"},"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>\u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e<\/h1>\n<p style=\"text-align:center;\"><em><br \/>\n<strong>\u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e<\/strong> \u0938\u0902\u0916\u094d\u092f\u093e \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 \u0915\u093e \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092a\u094d\u0930\u093e\u0930\u0902\u092d\u093f\u0915 \u092c\u093f\u0902\u0926\u0941 \u0939\u0948, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u092f\u0939 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u094b \u0938\u0902\u0930\u091a\u0928\u093e \u0935\u093e\u0932\u0947 \u090f\u0915 \u0924\u0902\u0924\u094d\u0930 \u092e\u0947\u0902 \u092c\u0926\u0932 \u0926\u0947\u0924\u0940 \u0939\u0948: \u0905\u092c \u0906\u092a \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u201c\u092e\u093e\u0924\u094d\u0930\u093e\u090f\u0901\u201d \u0928\u0939\u0940\u0902, \u092c\u0932\u094d\u0915\u093f \u0910\u0938\u0947 \u0918\u091f\u0915 \u092e\u093e\u0928\u0924\u0947 \u0939\u0948\u0902 \u091c\u094b \u090f\u0915-\u0926\u0942\u0938\u0930\u0947 \u0915\u0947 \u0938\u093e\u0925 \u092e\u0947\u0932 \u0916\u093e\u0924\u0947 \u0939\u0948\u0902 \u092f\u093e \u0928\u0939\u0940\u0902\u0964 \u090f\u0915 \u0939\u0940 \u0938\u0902\u0930\u091a\u0928\u093e, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u0906\u092a \u0938\u0930\u0932\u0940\u0915\u0930\u0923 \u0914\u0930 \u0917\u0941\u0923\u0928\u0916\u0902\u0921\u0928 \u0915\u0947 \u092e\u093e\u0928\u0926\u0902\u0921\u094b\u0902 \u0938\u0947 \u0932\u0947\u0915\u0930 \u092f\u0942\u0915\u094d\u0932\u093f\u0921 \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0925\u094d\u092e \u091c\u0948\u0938\u0940 \u0915\u0941\u091b \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e\u0913\u0902 \u0915\u0947 \u092e\u0942\u0932 \u0924\u0915 \u0938\u092c \u0915\u0941\u091b \u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u091c\u094b \u092c\u0921\u093c\u0947 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0938\u093e\u0925 \u092d\u0940 \u0915\u0941\u091b \u0939\u0940 \u0938\u0947\u0915\u0902\u0921 \u092e\u0947\u0902 \u092e\u0939\u0924\u094d\u0924\u092e \u0938\u092e\u093e\u092a\u0935\u0930\u094d\u0924\u0915 \u0928\u093f\u0915\u093e\u0932\u0928\u093e \u0938\u0902\u092d\u0935 \u092c\u0928\u093e\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0915\u0947 \u0905\u0924\u093f\u0930\u093f\u0915\u094d\u0924, \u092f\u0939 \u0909\u0928 \u0935\u093f\u091a\u093e\u0930\u094b\u0902 \u0915\u093e \u0924\u0915\u0928\u0940\u0915\u0940 \u0906\u0927\u093e\u0930 \u0939\u0948 \u091c\u094b \u0905\u0928\u0941\u092a\u094d\u0930\u092f\u0941\u0915\u094d\u0924 \u0917\u0923\u093f\u0924 \u0914\u0930 \u0938\u0902\u0917\u0923\u0928 \u092e\u0947\u0902 \u092c\u093e\u0930-\u092c\u093e\u0930 \u092a\u094d\u0930\u0915\u091f \u0939\u094b\u0924\u0947 \u0939\u0948\u0902: \u0938\u0930\u094d\u0935\u093e\u0902\u0917\u0938\u092e\u0924\u093e\u090f\u0901, \u092e\u0949\u0921\u094d\u092f\u0942\u0932\u0930 \u0905\u0902\u0915\u0917\u0923\u093f\u0924, \u0938\u0924\u094d\u092f\u093e\u092a\u0928, \u0915\u094b\u0921, \u0914\u0930 (\u0906\u0917\u0947 \u091a\u0932\u0915\u0930) \u0915\u094d\u0930\u093f\u092a\u094d\u091f\u094b\u0917\u094d\u0930\u093e\u092b\u0940\u0964 \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u092e\u0947\u0902 \u0926\u0915\u094d\u0937 \u0939\u094b\u0928\u093e, \u092e\u0942\u0932\u0924\u0903, \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u092e\u0947\u0902 \u0905\u0926\u0943\u0936\u094d\u092f \u092a\u094d\u0930\u0924\u093f\u0930\u0942\u092a\u094b\u0902 \u0915\u093e \u092a\u0924\u093e \u0932\u0917\u093e\u0928\u093e \u0914\u0930 \u0909\u0928\u094d\u0939\u0947\u0902 \u0910\u0938\u0947 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e\u0913\u0902 \u092e\u0947\u0902 \u0930\u0942\u092a\u093e\u0902\u0924\u0930\u093f\u0924 \u0915\u0930\u0928\u093e \u0938\u0940\u0916\u0928\u093e \u0939\u0948 \u091c\u094b \u0939\u092e\u0947\u0936\u093e \u0915\u093e\u092e \u0915\u0930\u0924\u0940 \u0939\u0948\u0902\u0964<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><b>\u0905\u0927\u093f\u0917\u092e \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f<\/b><br \/>\n\u0907\u0938 \u0905\u092aunte \u0915\u0947 \u0905\u0902\u0924 \u092e\u0947\u0902 \u091b\u093e\u0924\u094d\u0930 \u0938\u0915\u094d\u0937\u092e \u0939\u094b\u0917\u093e:\n<\/p>\n<ol>\n<li>\u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0938\u0902\u092c\u0902\u0927 \u0915\u094b <strong>\u0938\u092e\u091d\u0928\u093e<\/strong>\u0964<\/li>\n<li>\u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0914\u0930 \u0909\u0938\u0915\u0947 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b <strong>\u0938\u092e\u091d\u0928\u093e<\/strong>\u0964<\/li>\n<li>\u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0938\u0947 \u0938\u0902\u092c\u0902\u0927\u093f\u0924 \u092a\u0930\u093f\u0923\u093e\u092e\u094b\u0902 \u0914\u0930 \u092a\u094d\u0930\u092e\u0947\u092f\u094b\u0902 \u0915\u0940 \u0917\u0923\u093f\u0924\u0940\u092f \u0938\u093f\u0926\u094d\u0927\u093f\u092f\u093e\u0901 <strong>\u0935\u093f\u0915\u0938\u093f\u0924 \u0915\u0930\u0928\u093e<\/strong>\u0964<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><b><u>\u0935\u093f\u0937\u092f-\u0938\u0942\u091a\u0940<\/u><\/b><br \/>\n<a href=\"#1\">\u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e<\/a><br \/>\n<a href=\"#2\">\u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u0947 \u092e\u094c\u0932\u093f\u0915 \u0917\u0941\u0923<\/a><br \/>\n<a href=\"#3\">\u092a\u094d\u0930\u0938\u094d\u0924\u093e\u0935\u093f\u0924 \u0905\u092d\u094d\u092f\u093e\u0938<\/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>\u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e<\/h2>\n<p style=\"text-align: justify;\">\n\u201c<em>a, b \u0915\u094b \u0935\u093f\u092d\u093e\u091c\u093f\u0924 \u0915\u0930\u0924\u093e \u0939\u0948<\/em>\u201d \u0915\u093e \u0905\u0928\u094c\u092a\u091a\u093e\u0930\u093f\u0915 \u0935\u093f\u091a\u093e\u0930 \u0924\u092c \u0938\u091f\u0940\u0915 \u0939\u094b \u091c\u093e\u0924\u093e \u0939\u0948 \u091c\u092c \u0939\u092e \u0909\u0938\u0947 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a \u090f\u0915 \u0938\u0902\u092c\u0902\u0927 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964 \u0939\u092e \u0915\u0939\u0947\u0902\u0917\u0947 \u0915\u093f \u090f\u0915 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u090f\u0915 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u0915\u094b <strong>\u0935\u093f\u092d\u093e\u091c\u093f\u0924<\/strong> \u0915\u0930\u0924\u093e \u0939\u0948, \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u0915\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u0915\u0947 \u090f\u0915 \u0920\u0940\u0915-\u0920\u0940\u0915 \u0917\u0941\u0923\u091c \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0932\u093f\u0916\u093e \u091c\u093e \u0938\u0915\u0947\u0964 \u092f\u0939 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u092a\u0942\u0930\u0947 \u0905\u092aunte \u0915\u093e \u0906\u0927\u093e\u0930 \u0939\u0948, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u092f\u0939 \u201c\u0920\u0940\u0915 \u0938\u0947 \u092b\u093f\u091f \u0939\u094b\u0924\u093e \u0939\u0948\u201d \u091c\u0948\u0938\u0940 \u092c\u093e\u0924\u094b\u0902 \u0915\u094b \u090f\u0915 \u0938\u0924\u094d\u092f\u093e\u092a\u0928\u0940\u092f \u092e\u093e\u0928\u0926\u0902\u0921 \u092e\u0947\u0902 \u092c\u0926\u0932 \u0926\u0947\u0924\u0940 \u0939\u0948\u0964\n<\/p>\n<p style=\"text-align: justify;\">\n<strong>\u092a\u0930\u093f\u092d\u093e\u0937\u093e.<\/strong> \u092e\u093e\u0928 \u0932\u0947\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}<\/span><\/span> \u091c\u0939\u093e\u0901 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>\u0964 \u0939\u092e \u0915\u0939\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u0915\u094b \u0935\u093f\u092d\u093e\u091c\u093f\u0924 \u0915\u0930\u0924\u093e \u0939\u0948, \u0914\u0930 \u0932\u093f\u0916\u0924\u0947 \u0939\u0948\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, \u092f\u0926\u093f \u0914\u0930 \u0915\u0947\u0935\u0932 \u092f\u0926\u093f \u0915\u094b\u0908 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> \u0910\u0938\u093e \u0939\u094b \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=ka<\/span><\/span>\u0964 \u0905\u0928\u094d\u092f\u0925\u093e, \u0939\u092e \u0932\u093f\u0916\u0924\u0947 \u0939\u0948\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\nmid b<\/span><\/span>\u0964\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\u0907\u0938 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u092e\u0947\u0902, \u0938\u0902\u0916\u094d\u092f\u093e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> \u0915\u094b \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0938\u0947 \u0938\u0902\u092c\u0926\u094d\u0927 <strong>\u092d\u093e\u0917\u092b\u0932<\/strong> (\u092f\u093e \u0917\u0941\u0923\u0915) \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f, \u092f\u0939 \u0915\u0939\u0928\u093e \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">6\\mid 42<\/span><\/span> \u0907\u0938\u0915\u0947 \u0938\u092e\u0924\u0941\u0932\u094d\u092f \u0939\u0948 \u0915\u093f \u0915\u094b\u0908 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> \u0910\u0938\u093e \u092e\u094c\u091c\u0942\u0926 \u0939\u0948 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">42=6k<\/span><\/span>; \u0907\u0938 \u0938\u094d\u0925\u093f\u0924\u093f \u092e\u0947\u0902, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=7<\/span><\/span> \u0932\u0947\u0928\u093e \u092a\u0930\u094d\u092f\u093e\u092a\u094d\u0924 \u0939\u0948\u0964\n<\/p>\n<h3>\u092f\u0939 \u0927\u094d\u092f\u093e\u0928 \u092e\u0947\u0902 \u0930\u0916\u0928\u093e \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0939\u0948<\/h3>\n<ul>\n<li>\n    \u0936\u0930\u094d\u0924 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span> \u0905\u0928\u093f\u0935\u093e\u0930\u094d\u092f \u0939\u0948\u0964 \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u092f\u0926\u093f \u0939\u092e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=0<\/span><\/span> \u0915\u094b \u0905\u0928\u0941\u092e\u0924\u093f \u0926\u0947\u0928\u0947 \u0915\u093e \u092a\u094d\u0930\u092f\u093e\u0938 \u0915\u0930\u0947\u0902, \u0924\u094b \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u0940 \u0936\u0930\u094d\u0924 \u092f\u0939 \u092e\u093e\u0901\u0917\u0947\u0917\u0940 \u0915\u093f \u0915\u094b\u0908 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> \u0910\u0938\u093e \u0939\u094b \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0\\cdot k<\/span><\/span>\u0964 \u092a\u0930\u0902\u0924\u0941 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\cdot k=0<\/span><\/span> \u0939\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> \u0915\u0947 \u0932\u093f\u090f \u0939\u094b\u0924\u093e \u0939\u0948, \u0907\u0938\u0932\u093f\u090f \u090f\u0915\u092e\u093e\u0924\u094d\u0930 \u0938\u0902\u092d\u093e\u0935\u0928\u093e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span> \u0939\u0940 \u0939\u094b\u0917\u0940\u0964 \u0909\u0938 \u0938\u094d\u0925\u093f\u0924\u093f \u092e\u0947\u0902, \u0938\u0902\u092c\u0902\u0927 \u0926\u094d\u0935\u093e\u0930\u093e \u201c\u0928\u093f\u0930\u094d\u0927\u093e\u0930\u093f\u0924\u201d \u0915\u094b\u0908 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> \u0928\u0939\u0940\u0902 \u0939\u094b\u0917\u093e, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0915\u094b\u0908 \u092d\u0940 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=0\\cdot k<\/span><\/span> \u0915\u094b \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0915\u0930\u0924\u093e \u0939\u0948\u0964 \u0926\u0942\u0938\u0930\u0947 \u0936\u092c\u094d\u0926\u094b\u0902 \u092e\u0947\u0902, \u0905\u0928\u094c\u092a\u091a\u093e\u0930\u093f\u0915 \u0905\u092d\u093f\u0935\u094d\u092f\u0915\u094d\u0924\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=b\/a<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0\/0<\/span><\/span> \u092c\u0928 \u091c\u093e\u0924\u0940 \u0939\u0948, \u091c\u094b \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0928\u0939\u0940\u0902 \u0939\u0948\u0964 \u0907\u0938 \u0935\u093f\u0915\u0943\u0924\u093f \u0938\u0947 \u092c\u091a\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f (\u091c\u0939\u093e\u0901 \u092d\u093e\u0917\u092b\u0932 \u0915\u0940 \u0927\u093e\u0930\u0923\u093e \u0905\u0930\u094d\u0925\u092a\u0942\u0930\u094d\u0923 \u0928\u0939\u0940\u0902 \u0930\u0939\u0924\u0940), <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span> \u0915\u0940 \u092e\u093e\u0901\u0917 \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948\u0964 \u0907\u0938\u0940 \u0915\u093e\u0930\u0923 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\mid b<\/span><\/span> \u0938\u0902\u092c\u0902\u0927 \u0915\u094b \u0935\u0948\u0927 \u0928\u0939\u0940\u0902 \u092e\u093e\u0928\u093e \u091c\u093e\u0924\u093e\u0964\n<\/li>\n<li>\n        \u0907\u0938\u0915\u0947 \u0935\u093f\u092a\u0930\u0940\u0924, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid 0<\/span><\/span> \u0939\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{Z}<\/span><\/span> \u0915\u0947 \u0932\u093f\u090f \u0938\u0924\u094d\u092f \u0939\u0948 \u091c\u0939\u093e\u0901 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>, \u0915\u094d\u092f\u094b\u0902\u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span> \u0932\u0947\u0928\u093e \u092a\u0930\u094d\u092f\u093e\u092a\u094d\u0924 \u0939\u0948 \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=a\\cdot 0<\/span><\/span> \u0938\u0924\u094d\u092f \u0939\u094b \u091c\u093e\u0924\u093e \u0939\u0948\u0964\n    <\/li>\n<\/ul>\n<p style=\"text-align: justify;\">\n\u0907\u0938 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0938\u0947 \u090f\u0915 \u0938\u092e\u0924\u0941\u0932\u094d\u092f\u0924\u093e \u0928\u093f\u0915\u0932\u0924\u0940 \u0939\u0948 \u091c\u093f\u0938\u0915\u093e \u0939\u092e \u092c\u093e\u0930-\u092c\u093e\u0930 \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0947\u0902\u0917\u0947: \u092f\u0939 \u0915\u0939\u0928\u093e \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> \u0909\u0938\u0940 \u0915\u0947 \u092c\u0930\u093e\u092c\u0930 \u0939\u0948 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u0915\u0947 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0917\u0941\u0923\u091c\u094b\u0902 \u0915\u0947 \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0915\u093e \u0938\u0926\u0938\u094d\u092f \u0939\u0948, \u0905\u0930\u094d\u0925\u093e\u0924 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\in a\\mathbb{Z}<\/span><\/span>, \u091c\u0939\u093e\u0901 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mathbb{Z}=\\{ak:\\,k\\in\\mathbb{Z}\\}<\/span><\/span>\u0964 \u0907\u0938\u0947 \u0932\u093f\u0916\u0928\u0947 \u0915\u093e \u092f\u0939 \u0924\u0930\u0940\u0915\u093e \u0907\u0938 \u092c\u093e\u0924 \u092a\u0930 \u092c\u0932 \u0926\u0947\u0924\u093e \u0939\u0948 \u0915\u093f \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u094b\u0908 \u201c\u091a\u093e\u0932\u201d \u0928\u0939\u0940\u0902, \u092c\u0932\u094d\u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{Z}<\/span><\/span> \u0915\u0947 \u092d\u0940\u0924\u0930 \u0905\u0924\u094d\u092f\u0902\u0924 \u0938\u0902\u0930\u091a\u093f\u0924 \u0909\u092a\u0938\u092e\u0941\u091a\u094d\u091a\u092f\u094b\u0902 \u0915\u093e \u0935\u0930\u094d\u0923\u0928 \u0915\u0930\u0928\u0947 \u0915\u093e \u090f\u0915 \u0909\u092a\u093e\u092f \u0939\u0948\u0964<\/p>\n<p><a name=\"2\"><\/a><\/br><\/p>\n<h2>\u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u0947 \u092e\u0942\u0932\u092d\u0942\u0924 \u0917\u0941\u0923<\/h2>\n<ul>\n<li><strong>\u092a\u0930\u093e\u0935\u0930\u094d\u0924\u093f\u0924\u093e:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid a<\/span><\/span>.<br \/>\n<u>\u092a\u094d\u0930\u092e\u093e\u0923<\/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{; $\\mathbb{Z}$ \u092e\u0947\u0902 \u0917\u0941\u0923\u0928\u093e\u0924\u094d\u092e\u0915 \u090f\u0915\u0915}\\\\\n\n(2)&amp;\\vdash(\\exists k \\in \\mathbb{Z})(a=ka) &amp;\\text{; (1) \u0938\u0947 \u0905\u0938\u094d\u0924\u093f\u0924\u094d\u0935\u093e\u0924\u094d\u092e\u0915-\u092a\u0930\u093f\u091a\u092f}\\\\\n\n(3) &amp;\\vdash a \\mid a &amp;\\text{; \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e (2)} \\\\\n\n&amp;\\blacksquare &amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<li><strong>\u0938\u0902\u0915\u094d\u0930\u093e\u092e\u0915\u0924\u093e:<\/strong> \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\mid c<\/span><\/span>, \u0924\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>\u0964\n<p><u>\u092a\u094d\u0930\u092e\u093e\u0923<\/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{; \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e, \u092a\u0942\u0930\u094d\u0935\u0927\u093e\u0930\u0923\u093e}\\\\\n\n(2)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_2\\in\\mathbb{Z})(c=k_2b)  &amp;\\text{; \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e, \u092a\u0942\u0930\u094d\u0935\u0927\u093e\u0930\u0923\u093e}\\\\\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{; $\\exists$-\u0938\u0902\u0915\u0941\u091a\u0928(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{; (3) \u0938\u0947}\\\\\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{; (4) \u0938\u0947}\\\\\n\n&amp;\\text{\u092e\u093e\u0924\u094d\u0930\u0915 \u0915\u0947 \u092d\u0940\u0924\u0930 \u092c\u0940\u091c\u0917\u0923\u093f\u0924}&amp; \\\\\n\n(6)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k\\in\\mathbb{Z})( c=ka)  &amp;\\text{; (5) \u0938\u0947}\\\\\n\n&amp;\\text{\u0917\u0941\u0923\u0928 \u0915\u0947 \u0932\u093f\u090f $\\mathbb{Z}$ \u0915\u0940 \u092c\u0902\u0926\u0924\u093e}&amp; \\\\\n\n(7)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash a\\mid c  &amp;\\text{; \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e (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>\u092f\u094b\u0917 \u0914\u0930 \u0918\u091f\u093e\u0935 \u0915\u0947 \u0938\u093e\u0925 \u0938\u0902\u0917\u0924\u0924\u093e:<\/strong> \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>, \u0924\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (b+c)<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (b-c)<\/span><\/span>\u0964<br \/>\n<u>\u092a\u094d\u0930\u092e\u093e\u0923<\/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{; \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e, \u092a\u0942\u0930\u094d\u0935\u0927\u093e\u0930\u0923\u093e}\\\\\n\n(2)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_2 \\in \\mathbb{Z})(c=k_2 a) &amp;\\text{; \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e, \u092a\u0942\u0930\u094d\u0935\u0927\u093e\u0930\u0923\u093e}\\\\\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{; $\\exists$-\u0938\u0902\u0915\u0941\u091a\u0928(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{; (3) \u0938\u0947}\\\\\n\n&amp;\\text{\u092e\u093e\u0924\u094d\u0930\u0915 \u0915\u0947 \u092d\u0940\u0924\u0930 \u092c\u0940\u091c\u0917\u0923\u093f\u0924.}&amp; \\\\\n\n(5)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k \\in \\mathbb{Z})(b+c= ka) &amp;\\text{; (4) \u0938\u0947}\\\\\n\n&amp;\\text{\u092f\u094b\u0917 \u0915\u0947 \u0932\u093f\u090f $\\mathbb{Z}$ \u0915\u0940 \u092c\u0902\u0926\u0924\u093e.}&amp; \\\\\n\n(6)&amp;\\{a\\mid b, a\\mid c\\}\\vdash a\\mid (b+c) &amp;\\text{; \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e (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{; (3) \u0938\u0947}\\\\\n\n&amp;\\text{\u092e\u093e\u0924\u094d\u0930\u0915 \u0915\u0947 \u092d\u0940\u0924\u0930 \u092c\u0940\u091c\u0917\u0923\u093f\u0924.}&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{; (8) \u0938\u0947}\\\\\n\n&amp;\\text{\u0918\u091f\u093e\u0935 \u0915\u0947 \u0932\u093f\u090f $\\mathbb{Z}$ \u0915\u0940 \u092c\u0902\u0926\u0924\u093e.}&amp; \\\\\n\n(10)&amp;\\{a\\mid b, a\\mid c\\}\\vdash a\\mid (b-c) &amp;\\text{; \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e (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{;$\\wedge$-\u092a\u0930\u093f\u0923\u093e\u092e \u092e\u0947\u0902 \u092a\u0930\u093f\u091a\u092f(7,11) }\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}<\/span>\n<\/li>\n<li><strong>\u0917\u0941\u0923\u0928 \u0915\u0947 \u0938\u093e\u0925 \u0938\u0902\u0917\u0924\u0924\u093e:<\/strong> \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, \u0924\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (bc)<\/span><\/span> \u0938\u092d\u0940 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in\\mathbb{Z}<\/span><\/span> \u0915\u0947 \u0932\u093f\u090f\u0964<br \/>\n<u>\u092a\u094d\u0930\u092e\u093e\u0923<\/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{; \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e, \u092a\u0942\u0930\u094d\u0935\u0927\u093e\u0930\u0923\u093e}\\\\\n\n(2)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (\\exists k\\in\\mathbb{Z})(cb=cka) &amp;\\text{; (1) \u0938\u0947, $\\forall$-\u092a\u0930\u093f\u091a\u092f (c \u092e\u0928\u092e\u093e\u0928\u093e)}\\\\\n\n&amp;\\text{ \u092e\u0947\u0902 \u092c\u0940\u091c\u0917\u0923\u093f\u0924 }\\mathbb{Z}\\text{ \u0905\u0938\u094d\u0924\u093f\u0924\u094d\u0935\u093e\u0924\u094d\u092e\u0915 \u092e\u093e\u0924\u094d\u0930\u0915 \u0915\u0947 \u092d\u0940\u0924\u0930.}&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{; (2) \u0938\u0947, \u092c\u0902\u0926\u0924\u093e: }\\overline{k}=ck\\\\\n\n(4)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (a \\mid cb) &amp;\\text{; \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e (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<p>\u092f\u093e\u0926 \u0930\u0916\u0947\u0902 \u0915\u093f \\text{ } \u091f\u0948\u0917\u094b\u0902 \u0915\u0947 \u092d\u0940\u0924\u0930 \u0915\u0947 \u092a\u093e\u0920\u094b\u0902 \u0915\u093e \u0905\u0928\u0941\u0935\u093e\u0926 \u0915\u0930\u0947\u0902\u0964\n<\/ul>\n<h3><b>\u092a\u094d\u0930\u092e\u0947\u092f:<\/b> \u092d\u093e\u091c\u0915 \u0915\u0940 \u0938\u0940\u092e\u093e<\/h3>\n<p style=\"text-align: justify;\">\n\u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\neq 0<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"> <span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, \u0924\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|\\le |b|<\/span><\/span>\u0964\n<\/p>\n<p><b>\u092a\u094d\u0930\u092e\u093e\u0923:<\/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{; \u092a\u0942\u0930\u094d\u0935\u0927\u093e\u0930\u0923\u093e} \\\\\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{; \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e, \u092a\u0942\u0930\u094d\u0935\u0927\u093e\u0930\u0923\u093e} \\\\\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{; \u092a\u0930\u092e \u092e\u093e\u0928 \u0915\u093e \u0917\u0941\u0923, (2) \u0938\u0947} \\\\\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{; (1,3) \u0938\u0947} \\\\\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{; (4) \u0938\u0947, \u092f\u0926\u093f }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{; (5) \u0938\u0947} \\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<p><a name=\"3\"><\/a><\/br><\/p>\n<h2>\u092a\u094d\u0930\u0938\u094d\u0924\u093e\u0935\u093f\u0924 \u0905\u092d\u094d\u092f\u093e\u0938<\/h2>\n<ol>\n<li>\u0926\u093f\u0916\u093e\u090f\u0901 \u0915\u093f \u092a\u094d\u0930\u092e\u0947\u092f \u00ab\u092d\u093e\u091c\u0915 \u0915\u0940 \u0938\u0940\u092e\u093e\u00bb \u0905\u0928\u093f\u0935\u093e\u0930\u094d\u092f \u0930\u0942\u092a \u0938\u0947 \u0938\u0924\u094d\u092f \u0928\u0939\u0940\u0902 \u0939\u094b\u0924\u093e \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span><\/li>\n<li>\u090f\u0915 \u0938\u092e\u0941\u091a\u094d\u091a\u092f <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u0924\u0925\u093e \u0909\u0938 \u092a\u0930 \u090f\u0915 \u0938\u0902\u092c\u0902\u0927 <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0930\u0947\u0902\u0964 \u092f\u0926\u093f \u0924\u0924\u094d\u0935 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x,y\\in A<\/span> \u0910\u0938\u0947 \u0939\u094b\u0902 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u0915\u093e <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> \u0938\u0947 <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u0938\u0902\u092c\u0902\u0927 \u0939\u094b, \u0924\u094b \u0907\u0938\u0947 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\rho y<\/span><\/span><\/span> \u0932\u093f\u0916\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0938\u0902\u092c\u0902\u0927 <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> \u0915\u094b <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u092a\u0930 <strong>\u0906\u0902\u0936\u093f\u0915 \u0915\u094d\u0930\u092e<\/strong> \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u092f\u0926\u093f:\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>\u0964<\/p>\n<p>\u0938\u093f\u0926\u094d\u0927 \u0915\u0940\u091c\u093f\u090f \u0915\u093f \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0915\u093e \u0938\u0902\u092c\u0902\u0927 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u092a\u0930 \u090f\u0915 \u0906\u0902\u0936\u093f\u0915 \u0915\u094d\u0930\u092e \u0938\u0902\u092c\u0902\u0927 \u0939\u0948\u0964<\/li>\n<li>\u092a\u094d\u0930\u0947\u0930\u0923 \u0926\u094d\u0935\u093e\u0930\u093e \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f \u092f\u0926\u093f <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>, \u0924\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid \\sum_{i=1}^n b_i x_i<\/span><\/span> \u0915\u093f\u0938\u0940 \u092d\u0940 \u0938\u092e\u0941\u091a\u094d\u091a\u092f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{x_i\\}_{i=1}^n \\subset \\mathbb{Z}<\/span><\/span> \u0915\u0947 \u0932\u093f\u090f\u0964 \u0907\u0938\u0915\u0947 \u0905\u0924\u093f\u0930\u093f\u0915\u094d\u0924 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b_i<\/span><\/span>, \u091c\u0939\u093e\u0901 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i\\in \\{1,2,3,\\cdots, n\\}<\/span><\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> \u0915\u094b \u0909\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b_i<\/span><\/span> \u0915\u0947 \u0930\u0948\u0916\u093f\u0915 \u0938\u0902\u092f\u094b\u091c\u0928 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0932\u093f\u0916\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948, \u0924\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>\u0964 <\/li>\n<li>\u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>, \u0926\u093f\u0916\u093e\u090f\u0901 \u0915\u093f \u0938\u092e\u0941\u091a\u094d\u091a\u092f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{x\\;:\\; d\\mid a\\}<\/span><\/span> \u090f\u0915 \u0938\u0940\u092e\u093f\u0924 \u0938\u092e\u0941\u091a\u094d\u091a\u092f \u0939\u0948\u0964<\/li>\n<li>\u090f\u0915 \u0928\u093f\u092f\u0924 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\in\\mathbb{Z}^+<\/span><\/span> \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0930\u0947\u0902, \u0914\u0930 \u092e\u093e\u0928 \u0932\u0947\u0902\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>\u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902:<\/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\">\u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> \u0915\u0947 \u0924\u0924\u094d\u0924\u094d\u0935 \u0906\u0930\u094b\u0939\u0940 \u0915\u094d\u0930\u092e \u092e\u0947\u0902 \u0930\u0916\u0947 \u091c\u093e\u090f\u0901: <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>, \u0924\u094b \u0905\u0928\u0941\u0930\u0942\u092a \u0924\u0924\u094d\u0924\u094d\u0935 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\mid d_i<\/span><\/span> \u091c\u0939\u093e\u0901 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i \\in \\{1,2,\\cdots, t\\}<\/span><\/span> \u0905\u0935\u0930\u094b\u0939\u0940 \u0915\u094d\u0930\u092e \u092e\u0947\u0902 \u0939\u094b\u0902\u0917\u0947\u0964<\/li>\n<\/ol>\n<\/li>\n<li>\u092e\u093e\u0928 \u0932\u0947\u0902 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}^+<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ab=c<\/span><\/span>\u0964 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\min\\{a,b\\}\\le \\sqrt{c}<\/span><\/span>\u0964<\/li>\n<li>\u090f\u0915 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u0915\u094b \u0938\u092e \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2\\mid n<\/span><\/span>, \u0924\u0925\u093e \u0935\u093f\u0937\u092e \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2\\nmid n<\/span><\/span>\u0964 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f \u0928\u093f\u092e\u094d\u0928 \u0915\u0940 \u092f\u094b\u0917 \u0914\u0930 \u0905\u0902\u0924\u0930:\n<ol>\n<li type=\"a\">\u0926\u094b \u0938\u092e \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u092a\u0930\u093f\u0923\u093e\u092e \u090f\u0915 \u0938\u092e \u0938\u0902\u0916\u094d\u092f\u093e \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/li>\n<li type=\"a\">\u0926\u094b \u0935\u093f\u0937\u092e \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u092a\u0930\u093f\u0923\u093e\u092e \u090f\u0915 \u0938\u092e \u0938\u0902\u0916\u094d\u092f\u093e \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/li>\n<li type=\"a\">\u090f\u0915 \u0938\u092e \u0914\u0930 \u090f\u0915 \u0935\u093f\u0937\u092e \u0915\u093e \u092a\u0930\u093f\u0923\u093e\u092e \u090f\u0915 \u0935\u093f\u0937\u092e \u0938\u0902\u0916\u094d\u092f\u093e \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/li>\n<\/ol>\n<\/li>\n<li>\u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u090f\u0915 \u0935\u093f\u0937\u092e \u0938\u0902\u0916\u094d\u092f\u093e \u0939\u0948 \u091c\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\pm 1<\/span><\/span> \u0938\u0947 \u092d\u093f\u0928\u094d\u0928 \u0939\u0948, \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u0926\u094b \u0915\u094d\u0930\u092e\u093e\u0917\u0924 \u0938\u092e \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u0935\u093f\u092d\u093e\u091c\u093f\u0924 \u0928\u0939\u0940\u0902 \u0915\u0930 \u0938\u0915\u0924\u093e\u0964<\/li>\n<li>\u092e\u093e\u0928 \u0932\u0947\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b,n\\in\\mathbb{Z}<\/span><\/span> \u0910\u0938\u0947 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a-b|\\lt |n|<\/span><\/span>\u0964 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u0928 \u0924\u094b <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> \u0915\u094b \u0914\u0930 \u0928 \u0939\u0940 <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> \u0915\u094b \u0935\u093f\u092d\u093e\u091c\u093f\u0924 \u0915\u0930 \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/li>\n<li>\u092e\u093e\u0928 \u0932\u0947\u0902 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{Z}<\/span><\/span>\u0964 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f:\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>\u092e\u093e\u0928 \u0932\u0947\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{Z}<\/span><\/span> \u0924\u0925\u093e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\neq 0<\/span><\/span>\u0964 \u0926\u093f\u0916\u093e\u090f\u0901 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ac\\mid bc<\/span><\/span> \u0938\u0947 \u092f\u0939 \u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937 \u0928\u093f\u0915\u0932\u0924\u093e \u0939\u0948 \u0915\u093f <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>\u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0924\u093e \u0938\u0902\u0916\u094d\u092f\u093e \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 \u0915\u093e \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092a\u094d\u0930\u093e\u0930\u0902\u092d\u093f\u0915 \u092c\u093f\u0902\u0926\u0941 \u0939\u0948, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u092f\u0939 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u094b \u0938\u0902\u0930\u091a\u0928\u093e \u0935\u093e\u0932\u0947 \u090f\u0915 \u0924\u0902\u0924\u094d\u0930 \u092e\u0947\u0902 \u092c\u0926\u0932 \u0926\u0947\u0924\u0940 \u0939\u0948: \u0905\u092c \u0906\u092a \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u201c\u092e\u093e\u0924\u094d\u0930\u093e\u090f\u0901\u201d \u0928\u0939\u0940\u0902, \u092c\u0932\u094d\u0915\u093f \u0910\u0938\u0947 \u0918\u091f\u0915 \u092e\u093e\u0928\u0924\u0947 \u0939\u0948\u0902 \u091c\u094b \u090f\u0915-\u0926\u0942\u0938\u0930\u0947 \u0915\u0947 \u0938\u093e\u0925 \u092e\u0947\u0932 \u0916\u093e\u0924\u0947 \u0939\u0948\u0902 \u092f\u093e \u0928\u0939\u0940\u0902\u0964 \u090f\u0915 \u0939\u0940 \u0938\u0902\u0930\u091a\u0928\u093e, , \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u0906\u092a \u0938\u0930\u0932\u0940\u0915\u0930\u0923 \u0914\u0930 \u0917\u0941\u0923\u0928\u0916\u0902\u0921\u0928 \u0915\u0947 \u092e\u093e\u0928\u0926\u0902\u0921\u094b\u0902 \u0938\u0947 [&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":7,"footnotes":""},"categories":[577,1410],"tags":[],"class_list":["post-35678","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-577","category-1410"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>\u0915\u094d\u092f\u093e \u0939\u0948 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\u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u092a\u094d\u0930\u092e\u093e\u0923 \u0914\u0930 \u092a\u094d\u0930\u0938\u094d\u0924\u093e\u0935\u093f\u0924 \u0905\u092d\u094d\u092f\u093e\u0938\u0964\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/divisibilidad.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"8 minutos\" \/>\n<script type=\"application\/ld+json\" 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