{"id":35994,"date":"2026-01-10T18:40:43","date_gmt":"2026-01-10T18:40:43","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35994"},"modified":"2026-01-10T18:40:43","modified_gmt":"2026-01-10T18:40:43","slug":"%d8%ae%d9%88%d8%a7%d8%b1%d8%b2%d9%85%d9%8a%d8%a9-%d8%a7%d9%84%d9%82%d8%b3%d9%85%d8%a9-%d8%a7%d9%84%d8%a5%d9%82%d9%84%d9%8a%d8%af%d9%8a%d8%a9","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/ar\/%d8%ae%d9%88%d8%a7%d8%b1%d8%b2%d9%85%d9%8a%d8%a9-%d8%a7%d9%84%d9%82%d8%b3%d9%85%d8%a9-%d8%a7%d9%84%d8%a5%d9%82%d9%84%d9%8a%d8%af%d9%8a%d8%a9\/","title":{"rendered":"\u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629"},"content":{"rendered":"<style>\np, ul, ol{\n  text-align: justify;\n}\nh1{\n  text-align:center;\n  text-transform: uppercase;\n}\nh2{\n  text-align:center;\n  text-transform: uppercase;\n  font-size:24pt;\n}\nh3{\n  text-align: center;\n  text-transform: uppercase;\n  font-size: 24px !important;\n}\n<\/style>\n<h1>\u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629<\/h1>\n<p style=\"text-align:center;\"><em>\u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u062d\u0635\u0629 \u0633\u0646\u0637\u0648\u0651\u0631 <strong>\u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629<\/strong> \u0628\u0648\u0635\u0641\u0647\u0627 \u0627\u0644\u0645\u0628\u062f\u0623 \u0627\u0644\u0630\u064a \u064a\u0636\u0641\u064a \u0627\u0644\u0637\u0627\u0628\u0639 \u0627\u0644\u0631\u0633\u0645\u064a\u060c \u0639\u0644\u0649 \u0645\u0633\u062a\u0648\u0649 \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0635\u062d\u064a\u062d\u0629\u060c \u0639\u0644\u0649 \u0627\u0644\u062a\u0641\u0643\u064a\u0643 \u0627\u0644\u0648\u062d\u064a\u062f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span> \u0645\u0639 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>. \u064a\u064f\u0628\u0631\u0647\u064e\u0646 \u0623\u0648\u0644\u0627\u064b \u0639\u0644\u0649 <strong>\u0648\u062c\u0648\u062f<\/strong> \u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629 \u0648\u0627\u0644\u0628\u0627\u0642\u064a\u060c \u062b\u0645 \u0639\u0644\u0649 <strong>\u0648\u062d\u062f\u0627\u0646\u064a\u062a\u0647\u0645\u0627<\/strong>. \u0648\u0623\u062e\u064a\u0631\u0627\u064b\u060c \u064a\u064f\u0641\u0633\u0651\u064e\u0631 \u0645\u0639\u0646\u0649 \u0627\u0644\u0628\u0627\u0642\u064a\u060c \u0648\u062a\u064f\u0631\u0628\u0637 \u0627\u0644\u0646\u0638\u0631\u064a\u0629 \u0628\u0640<strong>\u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0637\u0648\u064a\u0644\u0629<\/strong> \u0628\u0648\u0635\u0641\u0647\u0627 \u0625\u062c\u0631\u0627\u0621\u064b \u062d\u0633\u0627\u0628\u064a\u0627\u064b\u060c \u0643\u0645\u0627 \u064a\u064f\u0633\u062a\u0628\u0642 \u0627\u0631\u062a\u0628\u0627\u0637\u0647\u0627 \u0627\u0644\u0637\u0628\u064a\u0639\u064a \u0628\u0627\u0644\u062d\u0633\u0627\u0628\u064a\u0627\u062a \u0627\u0644\u0645\u0639\u064a\u0627\u0631\u064a\u0629 \u0648\u062a\u0637\u0628\u064a\u0642\u0627\u062a\u0647\u0627 \u0627\u0644\u062d\u0627\u0633\u0648\u0628\u064a\u0629.<\/em><\/p>\n<p style=\"text-align:center;\"><strong>\u0623\u0647\u062f\u0627\u0641 \u0627\u0644\u062a\u0639\u0644\u0651\u0645<\/strong><\/p>\n<p>\u0639\u0646\u062f \u0625\u062a\u0645\u0627\u0645 \u0647\u0630\u0647 \u0627\u0644\u062d\u0635\u0629 \u0633\u064a\u0643\u0648\u0646 \u0627\u0644\u0637\u0627\u0644\u0628 \u0642\u0627\u062f\u0631\u0627\u064b \u0639\u0644\u0649:<\/p>\n<ol>\n<li><strong>\u062a\u062d\u062f\u064a\u062f<\/strong> \u0627\u0644\u0645\u0641\u0627\u0647\u064a\u0645 \u0648\u0627\u0644\u0623\u062f\u0648\u0627\u0631 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0635\u062d\u064a\u062d\u0629 (\u0627\u0644\u0645\u0642\u0633\u0648\u0645\u060c \u0627\u0644\u0645\u0642\u0633\u0648\u0645 \u0639\u0644\u064a\u0647\u060c \u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629\u060c \u0627\u0644\u0628\u0627\u0642\u064a) \u0648\u0645\u0641\u0647\u0648\u0645 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 \u0628\u0648\u0635\u0641\u0647\u0627 \u062d\u0627\u0644\u0629 \u062a\u0627\u0645\u0629.<\/li>\n<li><strong>\u0634\u0631\u062d<\/strong> \u0635\u064a\u0627\u063a\u0629 \u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629\/\u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629\u060c \u0628\u0645\u0627 \u0641\u064a \u0630\u0644\u0643 \u0627\u0644\u0634\u0631\u0648\u0637 \u0627\u0644\u062a\u064a \u062a\u062d\u062f\u062f \u0627\u0644\u0628\u0627\u0642\u064a (<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>) \u0648\u063a\u0631\u0636\u0647\u0627 \u0641\u064a \u062a\u062c\u0646\u0628 \u0627\u0644\u0627\u0644\u062a\u0628\u0627\u0633.<\/li>\n<li><strong>\u062a\u0637\u0628\u064a\u0642<\/strong> \u0627\u0644\u062a\u0641\u0643\u064a\u0643 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span> \u0644\u062a\u062d\u062f\u064a\u062f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> \u0641\u064a \u0623\u0645\u062b\u0644\u0629 \u0645\u0644\u0645\u0648\u0633\u0629\u060c \u0645\u0639 \u0627\u0644\u062a\u062d\u0642\u0642 \u0645\u0646 \u062d\u062f\u0651 \u0627\u0644\u0628\u0627\u0642\u064a.<\/li>\n<li><strong>\u062a\u062d\u0644\u064a\u0644<\/strong> \u0645\u0639\u0627\u0644\u062c\u0629 \u0627\u0644\u062d\u0627\u0644\u0627\u062a \u0628\u062d\u0633\u0628 \u0625\u0634\u0627\u0631\u0629 \u0627\u0644\u0645\u0642\u0633\u0648\u0645 \u0648\u0627\u0644\u0645\u0642\u0633\u0648\u0645 \u0639\u0644\u064a\u0647\u060c \u0648\u062a\u0628\u0631\u064a\u0631 \u0633\u0628\u0628 \u062d\u0641\u0627\u0638 \u0627\u0644\u0627\u0635\u0637\u0644\u0627\u062d \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a \u0639\u0644\u0649 \u0628\u0627\u0642\u064d \u063a\u064a\u0631 \u0633\u0627\u0644\u0628.<\/li>\n<li><strong>\u062a\u0646\u0641\u064a\u0630<\/strong> \u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0637\u0648\u064a\u0644\u0629 \u0628\u0648\u0635\u0641\u0647\u0627 \u0625\u062c\u0631\u0627\u0621\u064b \u0622\u0644\u064a\u0627\u064b \u0642\u0627\u0626\u0645\u0627\u064b \u0639\u0644\u0649 \u0627\u0644\u062a\u0645\u062b\u064a\u0644 \u0627\u0644\u0645\u0648\u0636\u0639\u064a \u0644\u0627\u0633\u062a\u062e\u0631\u0627\u062c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span>.<\/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\">\u0627\u0644\u0642\u0633\u0645\u0629 \u0648\u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629<\/a><br \/>\n<a href=\"#2\">\u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629<\/a><br \/>\n<a href=\"#3\">\u0628\u0631\u0647\u0627\u0646 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629<\/a><br \/>\n<a href=\"#4\">\u062a\u0641\u0633\u064a\u0631 \u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629 \u0648\u0627\u0644\u0628\u0627\u0642\u064a<\/a><br \/>\n<a href=\"#5\">\u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0637\u0648\u064a\u0644\u0629<\/a><br \/>\n<a href=\"#6\">\u0627\u0644\u062e\u0644\u0627\u0635\u0629<\/a><br \/>\n<a href=\"#7\">\u062a\u0645\u0627\u0631\u064a\u0646 \u0645\u0642\u062a\u0631\u062d\u0629 \u0648\u0645\u062d\u0644\u0648\u0644\u0629<\/a>\n<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u0627\u0644\u0642\u0633\u0645\u0629 \u0648\u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/nlER6VzrGwA?si=79gpsAzwc7wcWEVn\" 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><\/center><\/p>\n<p>\u0641\u064a \u0627\u0644\u062d\u0633\u0627\u0628\u060c \u062a\u0635\u0641 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u062d\u0627\u0644\u0629 \u00ab\u0627\u0644\u062a\u0627\u0645\u0629\u00bb: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> \u062a\u0639\u0646\u064a \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u064a\u064f\u0643\u062a\u0628 \u062a\u0645\u0627\u0645\u0627\u064b \u0628\u0648\u0635\u0641\u0647 \u0645\u0636\u0627\u0639\u0641\u0627\u064b \u0644\u0640<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>. \u0648\u0645\u0639 \u0630\u0644\u0643\u060c \u0648\u0639\u0644\u0649 \u0627\u0644\u0631\u063a\u0645 \u0645\u0646 \u0639\u062f\u0645 \u062a\u062d\u0642\u0642 \u0627\u0644\u0642\u0627\u0628\u0644\u064a\u0629 \u0644\u0644\u0642\u0633\u0645\u0629 \u062f\u0627\u0626\u0645\u0627\u064b \u0628\u064a\u0646 \u0623\u064a \u0639\u062f\u062f\u064a\u0646 \u0635\u062d\u064a\u062d\u064a\u0646\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u0645\u0639 \u0630\u0644\u0643 \u0623\u0646 \u0646\u062a\u0633\u0627\u0621\u0644 \u00ab\u0643\u0645 \u0645\u0631\u0629 \u064a\u062d\u062a\u0648\u064a \u0639\u062f\u062f\u064c \u0645\u0627 \u0639\u062f\u062f\u0627\u064b \u0622\u062e\u0631\u060c \u0648\u0625\u0630\u0627 \u0644\u0645 \u064a\u062d\u062a\u0648\u0647 \u062a\u0645\u0627\u0645\u0627\u064b\u060c \u0641\u0645\u0627 \u0627\u0644\u0630\u064a \u064a\u062a\u0628\u0642\u0649\u00bb. \u0641\u064a \u0647\u0630\u0627 \u0627\u0644\u0633\u064a\u0627\u0642 \u062a\u0638\u0647\u0631 <strong>\u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629<\/strong>\u060c \u0627\u0644\u062a\u064a \u062a\u0636\u0645\u0646 \u0623\u0646 \u0623\u064a \u0639\u062f\u062f \u0635\u062d\u064a\u062d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u064a\u0645\u0643\u0646 \u0643\u062a\u0627\u0628\u062a\u0647 \u0639\u0644\u0649 \u0623\u0646\u0647 \u0645\u0636\u0627\u0639\u0641 \u0644\u0640<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u0645\u0636\u0627\u0641\u0627\u064b \u0625\u0644\u064a\u0647 \u00ab\u0628\u0627\u0642\u064d\u00bb.<\/p>\n<p>\u0639\u0644\u0649 \u0633\u0628\u064a\u0644 \u0627\u0644\u0645\u062b\u0627\u0644\u060c \u0644\u0650\u0646\u0639\u062a\u0628\u0631 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=3<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=16<\/span><\/span>. \u0645\u0646 \u0627\u0644\u0648\u0627\u0636\u062d \u0623\u0646 3 \u0644\u0627 \u064a\u0642\u0633\u0645 16. \u0648\u0645\u0639 \u0630\u0644\u0643\u060c \u064a\u0645\u0643\u0646 \u0625\u064a\u062c\u0627\u062f \u062e\u0627\u0631\u062c \u0642\u0633\u0645\u0629 \u0648\u0628\u0627\u0642\u064d \u0644\u0639\u0645\u0644\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629\u060c \u0644\u0623\u0646<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">16 = 5\\cdot 3 + 1<\/span>\n<figure id=\"attachment_35775\" aria-describedby=\"caption-attachment-35775\" style=\"width: 1492px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3.jpg\" alt=\"Un grupo de 16 cajas han sido separadas en grupos de a 3, como resultado se obtienen 5 grupos de cajas y sobra 1\" width=\"1492\" height=\"230\" class=\"size-full wp-image-35775 lazyload\" \/><figcaption id=\"caption-attachment-35775\" class=\"wp-caption-text\"><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3.jpg\" alt=\"Un grupo de 16 cajas han sido separadas en grupos de a 3, como resultado se obtienen 5 grupos de cajas y sobra 1\" width=\"1492\" height=\"230\" class=\"size-full wp-image-35775 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3.jpg 1492w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3-300x46.jpg 300w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3-1024x158.jpg 1024w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3-768x118.jpg 768w\" sizes=\"(max-width: 1492px) 100vw, 1492px\" \/><\/noscript> \u062a\u0645 \u062a\u0642\u0633\u064a\u0645 \u0645\u062c\u0645\u0648\u0639\u0629 \u0645\u0646 16 \u0635\u0646\u062f\u0648\u0642\u0627\u064b \u0625\u0644\u0649 \u0645\u062c\u0645\u0648\u0639\u0627\u062a \u0645\u0646 3\u060c \u0641\u0643\u0627\u0646\u062a \u0627\u0644\u0646\u062a\u064a\u062c\u0629 5 \u0645\u062c\u0645\u0648\u0639\u0627\u062a \u0645\u0646 \u0627\u0644\u0635\u0646\u0627\u062f\u064a\u0642 \u0648\u0628\u0642\u064a \u0635\u0646\u062f\u0648\u0642 \u0648\u0627\u062d\u062f<\/figcaption><\/figure>\n<p>\u0648\u0628\u0646\u0627\u0621\u064b \u0639\u0644\u0649 \u0630\u0644\u0643\u060c \u0639\u0646\u062f \u0642\u0633\u0645\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=16<\/span><\/span> \u0639\u0644\u0649 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=3<\/span><\/span> (\u062d\u064a\u062b \u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u0647\u0648 <strong>\u0627\u0644\u0645\u0642\u0633\u0648\u0645<\/strong> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u0647\u0648 <strong>\u0627\u0644\u0645\u0642\u0633\u0648\u0645 \u0639\u0644\u064a\u0647<\/strong>)\u060c \u0648\u0627\u0644\u0645\u064f\u0645\u062b\u0651\u064e\u0644\u0629 \u0628\u0640<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">16\/3<\/span><\/span> \u0623\u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">16\\div 3<\/span><\/span>\u060c \u0646\u062d\u0635\u0644 \u0639\u0644\u0649 <strong>\u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=5<\/span><\/span> \u0648\u0639\u0644\u0649 <strong>\u0627\u0644\u0628\u0627\u0642\u064a<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=1<\/span><\/span>. \u0648\u0628\u0648\u062c\u0647 \u0639\u0627\u0645\u060c \u064a\u0624\u0643\u062f <em>\u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629<\/em> \u0623\u0646\u0647 \u0644\u0623\u064a \u0639\u062f\u062f\u064a\u0646 \u0635\u062d\u064a\u062d\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> (\u0627\u0644\u0645\u0642\u0633\u0648\u0645 \u0639\u0644\u064a\u0647) \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> (\u0627\u0644\u0645\u0642\u0633\u0648\u0645) \u0645\u0639 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>\u060c \u062a\u0648\u062c\u062f \u0623\u0639\u062f\u0627\u062f \u0635\u062d\u064a\u062d\u0629 \u0648\u062d\u064a\u062f\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> \u0628\u062d\u064a\u062b<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r,\\qquad 0\\le r&lt;|a|<\/span>\n<p>\u0641\u064a \u0647\u0630\u0627 \u0627\u0644\u0645\u062b\u0627\u0644\u060c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">16=3\\cdot 5+1<\/span><\/span>\u060c \u0648\u062a\u062a\u062d\u0642\u0642 \u0627\u0644\u0634\u0631\u0637\u064a\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span> \u0644\u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le 1&lt;3<\/span><\/span>. \u0648\u062a\u0645\u0646\u0639 \u0647\u0630\u0647 \u0627\u0644\u0635\u064a\u0627\u063a\u0629 \u062d\u062f\u0648\u062b \u0627\u0644\u0644\u0628\u0633 \u0639\u0646\u062f\u0645\u0627 \u064a\u0643\u0648\u0646 \u0627\u0644\u0645\u0642\u0633\u0648\u0645 \u0639\u0644\u064a\u0647 (\u0623\u0648 \u0627\u0644\u0645\u0642\u0633\u0648\u0645) \u0633\u0627\u0644\u0628\u0627\u064b\u060c \u0643\u0645\u0627 \u062a\u0636\u0645\u0646 \u0623\u0646 \u064a\u0643\u0648\u0646 \u0627\u0644\u0628\u0627\u0642\u064a \u062f\u0627\u0626\u0645\u0627\u064b \u063a\u064a\u0631 \u0633\u0627\u0644\u0628 \u0648\u0623\u0635\u063a\u0631 \u0645\u0646 \u0627\u0644\u0642\u064a\u0645\u0629 \u0627\u0644\u0645\u0637\u0644\u0642\u0629 \u0644\u0644\u0645\u0642\u0633\u0648\u0645 \u0639\u0644\u064a\u0647.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u0645\u0628\u0631\u0647\u0646\u0629 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629<\/h2>\n<p>\u0625\u0646 \u0646\u062a\u064a\u062c\u0629 \u062a\u0637\u0628\u064a\u0642 \u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629 \u0647\u064a \u0645\u0627 \u064a\u064f\u0639\u0631\u0641 \u0628\u0640<strong>\u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629<\/strong>\u060c \u0648\u0647\u064a \u062a\u0633\u062a\u0646\u062f \u0625\u0644\u0649 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0627\u0644\u0622\u062a\u064a\u0629.<\/p>\n<p><b>\u0645\u0628\u0631\u0647\u0646\u0629:<\/b> \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>. \u0639\u0646\u062f\u0626\u0630\u064d \u062a\u0648\u062c\u062f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> \u0648\u062d\u064a\u062f\u062a\u0627\u0646 \u0628\u062d\u064a\u062b<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nb=qa+r,\\qquad 0\\le r&lt;|a|.\n\n<\/span>\n<p>\u064a\u064f\u0633\u0645\u0651\u0649 \u0627\u0644\u0639\u062f\u062f \u0627\u0644\u0635\u062d\u064a\u062d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> <strong>\u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629<\/strong>\u060c \u0648\u064a\u064f\u0633\u0645\u0651\u0649 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> <strong>\u0627\u0644\u0628\u0627\u0642\u064a<\/strong> \u0644\u0642\u0633\u0645\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u0639\u0644\u0649 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u0628\u0631\u0647\u0627\u0646 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629<\/h2>\n<p>\u064a\u0646\u0642\u0633\u0645 \u0647\u0630\u0627 \u0627\u0644\u0628\u0631\u0647\u0627\u0646 \u0625\u0644\u0649 \u062c\u0632\u0623\u064a\u0646: \u064a\u064f\u0628\u0631\u0647\u064e\u0646 \u0623\u0648\u0644\u0627\u064b \u0639\u0644\u0649 \u0648\u062c\u0648\u062f \u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629 \u0648\u0627\u0644\u0628\u0627\u0642\u064a\u061b \u062b\u0645 \u064a\u064f\u0628\u064a\u0651\u064e\u0646 \u0623\u0646\u0647\u0645\u0627\u060c \u0628\u0639\u062f \u0625\u062b\u0628\u0627\u062a \u0648\u062c\u0648\u062f\u0647\u0645\u0627\u060c \u0648\u062d\u064a\u062f\u0627\u0646.<\/p>\n<h3>\u0627\u0644\u0648\u062c\u0648\u062f<\/h3>\n<p>\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>\u060c \u0648\u0644\u0652\u0646\u064f\u0639\u0631\u0651\u0641 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d=|a|<\/span><\/span> \u0628\u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d&gt;0<\/span><\/span>. \u0633\u0646\u0628\u0631\u0647\u0646 \u0623\u0648\u0644\u0627\u064b \u062d\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\ge 0<\/span><\/span>\u060c \u062d\u064a\u062b \u0633\u0646\u064f\u0638\u0647\u0631 \u0648\u062c\u0648\u062f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> \u064a\u062d\u0642\u0642\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq+r<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;d<\/span><\/span>. \u062b\u0645 \u0633\u0646\u062a\u0646\u0627\u0648\u0644 \u062d\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b&lt;0<\/span><\/span>. \u0648\u0641\u064a \u0627\u0644\u062e\u062a\u0627\u0645\u060c \u0633\u0646\u0633\u062a\u0628\u062f\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d<\/span><\/span> \u0628\u0640<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span> \u0644\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 \u0627\u0644\u0635\u064a\u063a\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span> \u0645\u0639 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>.<\/p>\n<p>\u0644\u0640<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\in\\mathbb{Z}<\/span><\/span> \u0645\u0639 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\ge 0<\/span><\/span>\u060c \u0646\u064f\u0639\u0631\u0651\u0641 \u0627\u0644\u0642\u0636\u064a\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(b)<\/span><\/span>: \u00ab\u062a\u0648\u062c\u062f \u0623\u0639\u062f\u0627\u062f \u0635\u062d\u064a\u062d\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> \u0628\u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq+r<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;d<\/span><\/span>\u00bb. \u0633\u0646\u0628\u0631\u0647\u0646 \u0628\u0627\u0644\u0627\u0633\u062a\u0642\u0631\u0627\u0621 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(b)<\/span><\/span> \u0635\u062d\u064a\u062d\u0629 \u0644\u0643\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\ge 0<\/span><\/span>.<\/p>\n<p><strong>\u062d\u0627\u0644\u0629 \u0627\u0644\u0623\u0633\u0627\u0633:<\/strong> \u0639\u0646\u062f\u0645\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span>\u060c \u0628\u0623\u062e\u0630 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=0<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=0<\/span><\/span> \u0646\u062d\u0635\u0644 \u0639\u0644\u0649 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=d\\cdot 0+0<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le 0&lt;d<\/span><\/span>. \u0648\u0639\u0644\u064a\u0647 \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(0)<\/span><\/span> \u0635\u062d\u064a\u062d\u0629.<\/p>\n<p><strong>\u0627\u0644\u062e\u0637\u0648\u0629 \u0627\u0644\u0627\u0633\u062a\u0642\u0631\u0627\u0626\u064a\u0629:<\/strong> \u0644\u064a\u0643\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\ge 0<\/span><\/span> \u0648\u0644\u0646\u0641\u0631\u0636 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(k)<\/span><\/span> \u0635\u062d\u064a\u062d\u0629. \u0625\u0630\u0646 \u062a\u0648\u062c\u062f \u0623\u0639\u062f\u0627\u062f \u0635\u062d\u064a\u062d\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> \u0628\u062d\u064a\u062b<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nk=dq+r,\\qquad 0\\le r&lt;d.\n\n<\/span>\n<p>\u0628\u0625\u0636\u0627\u0641\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span> \u0625\u0644\u0649 \u0627\u0644\u0637\u0631\u0641\u064a\u0646 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nk+1=dq+(r+1).\n\n<\/span>\n<p>\u0643\u0645\u0627 \u0623\u0646 \u0645\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;d<\/span><\/span> \u064a\u0646\u062a\u062c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1\\le d<\/span><\/span>. \u0648\u0628\u0646\u0627\u0621\u064b \u0639\u0644\u064a\u0647\u060c \u0644\u0627 \u064a\u0645\u0643\u0646 \u0625\u0644\u0627 \u0623\u0646 \u062a\u0642\u0639 \u0625\u062d\u062f\u0649 \u0627\u0644\u062d\u0627\u0644\u062a\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1&lt;d<\/span><\/span> \u0623\u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1=d<\/span><\/span>.<\/p>\n<p><b>\u0627\u0644\u062d\u0627\u0644\u0629 1:<\/b> \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1&lt;d<\/span><\/span>\u060c \u0646\u064f\u0639\u0631\u0651\u0641 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q&#039;=q<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r&#039;=r+1<\/span><\/span>. \u0648\u0639\u0646\u062f\u0626\u0630\u064d<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nk+1=dq&#039;+r&#039;,\\qquad 0\\le r&#039;&lt;d.\n\n<\/span>\n<p><b>\u0627\u0644\u062d\u0627\u0644\u0629 2:<\/b> \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1=d<\/span><\/span>\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k+1=dq+d=d(q+1)+0<\/span><\/span>. \u0646\u064f\u0639\u0631\u0651\u0641 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q&#039;=q+1<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r&#039;=0<\/span><\/span>\u060c \u0648\u064a\u0643\u0648\u0646<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nk+1=dq&#039;+r&#039;,\\qquad 0\\le r&#039;&lt;d.\n\n<\/span>\n<p>\u0641\u064a \u0643\u0644\u062a\u0627 \u0627\u0644\u062d\u0627\u0644\u062a\u064a\u0646 \u0628\u0646\u064a\u0646\u0627 \u0639\u062f\u062f\u064a\u0646 \u0635\u062d\u064a\u062d\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q&#039;,r&#039;<\/span><\/span> \u064a\u062d\u0642\u0642\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k+1=dq&#039;+r&#039;<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&#039;&lt;d<\/span><\/span>. \u0648\u0639\u0644\u064a\u0647 \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(k+1)<\/span><\/span> \u0635\u062d\u064a\u062d\u0629. \u0648\u0628\u0627\u0644\u0627\u0633\u062a\u0642\u0631\u0627\u0621 \u0646\u0633\u062a\u0646\u062a\u062c \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(b)<\/span><\/span> \u0635\u062d\u064a\u062d\u0629 \u0644\u0643\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\ge 0<\/span><\/span>.<\/p>\n<p>\u0648\u0627\u0644\u0622\u0646 \u0646\u0639\u062a\u0628\u0631 \u0627\u0644\u062d\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b&lt;0<\/span><\/span>. \u0639\u0646\u062f\u0626\u0630\u064d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-b&gt;0<\/span><\/span>. \u0628\u062a\u0637\u0628\u064a\u0642 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0627\u0644\u0633\u0627\u0628\u0642\u0629 \u0639\u0644\u0649 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-b<\/span><\/span>\u060c \u062a\u0648\u062c\u062f \u0623\u0639\u062f\u0627\u062f \u0635\u062d\u064a\u062d\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> \u0628\u062d\u064a\u062b<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n-b=dq+r,\\qquad 0\\le r&lt;d.\n\n<\/span>\n<p>\u0648\u0628\u0627\u0644\u0636\u0631\u0628 \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-1<\/span><\/span> \u0646\u062d\u0635\u0644 \u0639\u0644\u0649<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nb=-dq-r.\n\n<\/span>\n<p>\u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=0<\/span><\/span>\u060c \u0641\u064a\u0643\u0641\u064a \u0623\u062e\u0630 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_1=-q<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r_1=0<\/span><\/span>\u060c \u0641\u0646\u062d\u0635\u0644 \u0639\u0644\u0649 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq_1+r_1<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r_1&lt;d<\/span><\/span>.<\/p>\n<p>\u0648\u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r&gt;0<\/span><\/span>\u060c \u0646\u064f\u0639\u0631\u0651\u0641 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_1=-q-1<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r_1=d-r<\/span><\/span>. \u0648\u0628\u0645\u0627 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0&lt;r&lt;d<\/span><\/span>\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0&lt;d-r&lt;d<\/span><\/span>\u060c \u0623\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r_1&lt;d<\/span><\/span>. \u0625\u0636\u0627\u0641\u0629\u064b \u0625\u0644\u0649 \u0630\u0644\u0643\u060c<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\ndq_1+r_1=d(-q-1)+(d-r)=-dq-d+d-r=-dq-r=b.\n\n<\/span>\n<p>\u0648\u0628\u0646\u0627\u0621\u064b \u0639\u0644\u0649 \u0630\u0644\u0643\u060c \u0644\u0643\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\in\\mathbb{Z}<\/span><\/span> \u062a\u0648\u062c\u062f \u0623\u0639\u062f\u0627\u062f \u0635\u062d\u064a\u062d\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_1,r_1\\in\\mathbb{Z}<\/span><\/span> \u0628\u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq_1+r_1<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r_1&lt;d<\/span><\/span>.<\/p>\n<p>\u0648\u0623\u062e\u064a\u0631\u0627\u064b\u060c \u0644\u0646\u062a\u0630\u0643\u0651\u0631 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d=|a|<\/span><\/span>. \u0641\u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a&gt;0<\/span><\/span>\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d=a<\/span><\/span>\u060c \u0648\u062a\u064f\u0643\u062a\u0628 \u0627\u0644\u0645\u0633\u0627\u0648\u0627\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq_1+r_1<\/span><\/span> \u0639\u0644\u0649 \u0635\u0648\u0631\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=aq_1+r_1<\/span><\/span>\u060c \u0645\u0639 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r_1&lt;|a|<\/span><\/span>.<\/p>\n<p>\u0648\u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a&lt;0<\/span><\/span>\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d=-a<\/span><\/span>\u060c \u0648\u0639\u0646\u062f\u0626\u0630\u064d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq_1+r_1=(-a)q_1+r_1=a(-q_1)+r_1<\/span><\/span>. \u0648\u0628\u0650\u062a\u0639\u0631\u064a\u0641 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=-q_1<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=r_1<\/span><\/span> \u0646\u062d\u0635\u0644 \u0639\u0644\u0649<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nb=qa+r,\\qquad 0\\le r&lt;|a|.\n\n<\/span>\n<p>\u0648\u0628\u0630\u0644\u0643 \u062a\u062b\u0628\u062a \u0648\u062c\u0648\u062f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> \u0644\u0623\u064a <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>.<\/p>\n<h3>\u0627\u0644\u0648\u062d\u062f\u0627\u0646\u064a\u0629<\/h3>\n<p>\u0644\u0646\u0641\u062a\u0631\u0636 \u0648\u062c\u0648\u062f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,q&#039;,r,r&#039;\\in\\mathbb{Z}<\/span><\/span> \u0628\u062d\u064a\u062b<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nb=qa+r=q&#039;a+r&#039;,\\qquad 0\\le r,r&#039;&lt;|a|.\n\n<\/span>\n<p>\u0628\u0637\u0631\u062d \u0627\u0644\u0639\u0628\u0627\u0631\u062a\u064a\u0646 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\na(q-q&#039;)=r&#039;-r.\n\n<\/span>\n<p>\u0648\u0628\u0645\u0627 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r,r&#039;&lt;|a|<\/span><\/span>\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|r&#039;-r|&lt;|a|<\/span><\/span>. \u0623\u0645\u0627 \u0627\u0644\u0637\u0631\u0641 \u0627\u0644\u0623\u064a\u0633\u0631 \u0641\u0647\u0648 \u0645\u0636\u0627\u0639\u0641 \u0644\u0640<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>\u060c \u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a \u0647\u0648 \u0623\u064a\u0636\u0627\u064b \u0645\u0636\u0627\u0639\u0641 \u0644\u0640<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span>. \u0648\u0627\u0644\u0645\u0636\u0627\u0639\u0641 \u0627\u0644\u0635\u062d\u064a\u062d \u0627\u0644\u0648\u062d\u064a\u062f \u0644\u0640<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span> \u0627\u0644\u0630\u064a \u062a\u0643\u0648\u0646 \u0642\u064a\u0645\u062a\u0647 \u0627\u0644\u0645\u0637\u0644\u0642\u0629 \u0623\u0635\u063a\u0631 \u062a\u0645\u0627\u0645\u0627\u064b \u0645\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span> \u0647\u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span>\u061b \u0625\u0630 \u0625\u0646\u0647 \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a| m \\neq 0<\/span><\/span> \u0645\u0639 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m\\in\\mathbb{Z}<\/span><\/span>\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\big| |a| m \\big| \\ge |a| <\/span><\/span>. \u0648\u0639\u0644\u064a\u0647 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r&#039;-r=0<\/span><\/span>\u060c \u0623\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=r&#039;<\/span><\/span>\u060c \u0648\u0628\u0627\u0644\u062a\u0639\u0648\u064a\u0636 \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span> \u0646\u0633\u062a\u0646\u062a\u062c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=q&#039;<\/span><\/span>. \u0648\u0645\u0646 \u062b\u0645\u0651 \u064a\u0643\u0648\u0646 \u0627\u0644\u062a\u0641\u0643\u064a\u0643 \u0648\u062d\u064a\u062f\u0627\u064b.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>\u062a\u0641\u0633\u064a\u0631 \u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629 \u0648\u0627\u0644\u0628\u0627\u0642\u064a<\/h2>\n<ul>\n<li style=\"text-align: justify;\">\n    <strong>\u0645\u0627 \u0645\u0639\u0646\u0649 \u00ab\u0627\u0644\u0628\u0627\u0642\u064a\u00bb.<\/strong> \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span>\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=b-qa<\/span><\/span> \u0647\u0648 \u0645\u0627 \u00ab\u064a\u062a\u0628\u0642\u0649\u00bb \u0628\u0639\u062f \u0637\u0631\u062d \u0627\u0644\u0645\u0636\u0627\u0639\u0641 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">qa<\/span><\/span> \u0645\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span>. \u0648\u0628\u0639\u0628\u0627\u0631\u0629 \u0623\u062e\u0631\u0649\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> \u0647\u0648 \u0627\u0644\u0645\u062a\u0628\u0642\u064a \u0627\u0644\u0646\u0627\u062a\u062c \u0639\u0646 \u0645\u0648\u0627\u0621\u0645\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u0645\u0639 \u0634\u0628\u0643\u0629 \u0645\u0636\u0627\u0639\u0641\u0627\u062a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>.\n  <\/li>\n<li style=\"text-align: justify;\">\n    <strong>\u0644\u0645\u0627\u0630\u0627 \u064a\u064f\u0634\u062a\u0631\u0637<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>. \u0645\u0646 \u062f\u0648\u0646 \u0647\u0630\u0627 \u0627\u0644\u0634\u0631\u0637 \u0644\u0646 \u064a\u0643\u0648\u0646 \u0627\u0644\u0632\u0648\u062c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(q,r)<\/span><\/span> \u0648\u062d\u064a\u062f\u0627\u064b. \u0641\u0641\u064a \u0627\u0644\u0648\u0627\u0642\u0639\u060c \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span>\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=a(q+1)+(r-a)<\/span><\/span> \u0623\u064a\u0636\u0627\u064b. \u064a\u0641\u0631\u0636 \u0627\u0644\u0634\u0631\u0637 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span> \u0639\u0644\u0649 \u0627\u0644\u0628\u0627\u0642\u064a \u0623\u0646 \u064a\u0642\u0639 \u0636\u0645\u0646 \u0646\u0627\u0641\u0630\u0629 \u062b\u0627\u0628\u062a\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{0,1,2,\\dots,|a|-1\\}<\/span><\/span>\u060c \u0648\u0628\u0630\u0644\u0643 \u064a\u0645\u0646\u0639 \u00ab\u0625\u0639\u0627\u062f\u0629 \u062a\u0633\u0645\u064a\u0627\u062a\u00bb \u063a\u064a\u0631 \u0645\u0646\u062a\u0647\u064a\u0629 \u0644\u0644\u0639\u062f\u062f \u0646\u0641\u0633\u0647.\n  <\/li>\n<li style=\"text-align: justify;\">\n    <strong>\u0645\u0627\u0630\u0627 \u064a\u062d\u062f\u062b \u0625\u0630\u0627 \u0643\u0627\u0646<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> <strong>\u0633\u0627\u0644\u0628\u0627\u064b.<\/strong> \u062a\u0638\u0644 \u0627\u0644\u0645\u0628\u0631\u0647\u0646\u0629 \u0635\u062d\u064a\u062d\u0629 \u0645\u0646 \u062f\u0648\u0646 \u0623\u064a \u062a\u063a\u064a\u064a\u0631: \u064a\u0628\u0642\u0649 \u0627\u0644\u0628\u0627\u0642\u064a \u063a\u064a\u0631 \u0633\u0627\u0644\u0628 \u0648\u0645\u062d\u062f\u0648\u062f\u0627\u064b \u0628\u0640<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span>. \u0648\u0647\u0630\u0627 \u0623\u0645\u0631 \u0645\u0647\u0645 \u0644\u0623\u0646 \u0628\u0639\u0636 \u0644\u063a\u0627\u062a \u0627\u0644\u0628\u0631\u0645\u062c\u0629 \u062a\u0633\u062a\u062e\u062f\u0645 \u0627\u0644\u0642\u0637\u0639 \u0646\u062d\u0648 \u0627\u0644\u0635\u0641\u0631 \u0648\u0642\u062f \u062a\u064f\u0646\u062a\u062c \u0628\u0648\u0627\u0642\u064a \u0633\u0627\u0644\u0628\u0629\u060c \u0628\u064a\u0646\u0645\u0627 \u062a\u0639\u062a\u0645\u062f \u0627\u0644\u0631\u064a\u0627\u0636\u064a\u0627\u062a \u0627\u0644\u0627\u0635\u0637\u0644\u0627\u062d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span> \u0644\u064a\u0643\u0648\u0646 \u0627\u0644\u0628\u0627\u0642\u064a \u0645\u0645\u062b\u0644\u0627\u064b \u00ab\u0642\u064a\u0627\u0633\u064a\u0627\u064b\u00bb.<\/p>\n<p>    \u0625\u0636\u0627\u0641\u0629\u064b \u0625\u0644\u0649 \u0630\u0644\u0643\u060c \u0639\u0646\u062f\u0645\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a&gt;0<\/span><\/span>\u060c \u064a\u062a\u0637\u0627\u0628\u0642 \u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629 \u0645\u0639 \u062f\u0627\u0644\u0629 \u0627\u0644\u062c\u0632\u0621 \u0627\u0644\u0635\u062d\u064a\u062d:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nq=\\left\\lfloor \\frac{b}{a}\\right\\rfloor,\\qquad r=b-a\\left\\lfloor \\frac{b}{a}\\right\\rfloor.\n\n<\/span>\n<\/li>\n<\/ul>\n<p><a name=\"5\"><\/a><\/p>\n<h2>\u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0637\u0648\u064a\u0644\u0629<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ABw4QYdD6Pg?si=eJiA7P6cD7RdlYYN\" 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><\/center><\/p>\n<p>\u0627\u0646\u0637\u0644\u0627\u0642\u0627\u064b \u0645\u0646 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629 \u064a\u0645\u0643\u0646 \u062a\u0646\u0641\u064a\u0630 \u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0637\u0648\u064a\u0644\u0629\u060c \u0628\u0627\u0644\u0627\u0633\u062a\u0641\u0627\u062f\u0629 \u0645\u0646 \u0627\u0644\u062a\u0645\u062b\u064a\u0644 \u0627\u0644\u0645\u0648\u0636\u0639\u064a \u0644\u0644\u0623\u0639\u062f\u0627\u062f. \u0647\u0630\u0647 \u0627\u0644\u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0647\u064a \u062a\u0642\u0646\u064a\u0629 \u062d\u0633\u0627\u0628\u064a\u0629 \u062a\u062a\u064a\u062d \u0625\u064a\u062c\u0627\u062f \u0627\u0644\u0642\u064a\u0645 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> \u0628\u0633\u0631\u0639\u0629 \u0639\u0646\u062f\u0645\u0627 \u0646\u0631\u063a\u0628 \u0641\u064a \u062d\u0633\u0627\u0628 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\div a<\/span><\/span> \u0645\u0639 <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\">a\\neq 0<\/span><\/span>.<\/p>\n<p>\u0644\u0648\u0635\u0641\u0647\u0627\u060c \u0646\u0633\u062a\u0639\u0631\u0636 \u0623\u0648\u0644\u0627\u064b \u0628\u0639\u0636 \u0627\u0644\u0623\u0645\u062b\u0644\u0629 \u0627\u0644\u062a\u064a \u062a\u0648\u0636\u062d \u0627\u0644\u0639\u0645\u0644\u064a\u0629 \u0648\u0627\u0644\u062d\u0627\u0644\u0627\u062a \u0627\u0644\u0645\u0645\u0643\u0646\u0629 \u0623\u062b\u0646\u0627\u0621 \u062a\u0646\u0641\u064a\u0630 \u0627\u0644\u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629.<\/p>\n<ol>\n<li>\n<p>\u0644\u0646\u0641\u062a\u0631\u0636 \u0623\u0646\u0646\u0627 \u0646\u0631\u064a\u062f \u062d\u0633\u0627\u0628 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">57\\div 4<\/span><\/span>. \u0647\u0646\u0627 57 \u0647\u0648 <strong>\u0627\u0644\u0645\u0642\u0633\u0648\u0645<\/strong> \u06484 \u0647\u0648 <strong>\u0627\u0644\u0645\u0642\u0633\u0648\u0645 \u0639\u0644\u064a\u0647<\/strong>. \u0644\u062a\u062d\u0642\u064a\u0642 \u0630\u0644\u0643 \u0633\u0646\u062c\u0631\u064a \u0627\u0644\u062a\u0633\u0644\u0633\u0644 \u0627\u0644\u0622\u062a\u064a \u0645\u0646 \u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u062d\u0633\u0627\u0628\u064a\u0629:<\/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; \\color{red}57 \\div 4 =\\color{black}  \\\\ &amp; \\text{\u0643\u062a\u0627\u0628\u0629 \u0639\u0645\u0644\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629.} \\\\ \\\\\n\n(2) &amp; \\color{red}5&#039;\\color{black}7 \\div 4 =  \\\\ &amp; \\text{\u0641\u0635\u0644 \u0623\u0648\u0644 \u0628\u0627\u062f\u0626\u0629 (\u0645\u0646 \u0627\u0644\u064a\u0633\u0627\u0631) \u0645\u0646 \u0627\u0644\u0645\u0642\u0633\u0648\u0645 \u062a\u0643\u0648\u0646}\\\\\n\n&amp;\\text{\u0623\u0643\u0628\u0631 \u0645\u0646 \u0623\u0648 \u0645\u0633\u0627\u0648\u064a\u0629 \u0644\u0644\u0645\u0642\u0633\u0648\u0645 \u0639\u0644\u064a\u0647\u060c \u0648\u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u062d\u0627\u0644\u0629 5.} \\\\ \\\\\n\n(3) &amp; 5&#039;7 \\div 4 = \\color{red}1\\color{black} \\\\ &amp; \\text{\u0627\u0644\u062a\u0641\u0643\u064a\u0631 \u0641\u064a \u0623\u0643\u0628\u0631 \u0639\u062f\u062f \u0625\u0630\u0627 \u0636\u064f\u0631\u0628 \u0641\u064a $4$}\\\\\n\n&amp;\\text{\u0623\u0639\u0637\u0649 \u0646\u0627\u062a\u062c\u0627\u064b \u0623\u0635\u063a\u0631 \u0645\u0646 \u0623\u0648 \u0645\u0633\u0627\u0648\u064a\u0627\u064b \u0644\u0640$5$ \u0648\u0643\u062a\u0627\u0628\u062a\u0647 \u064a\u0645\u064a\u0646 \u0627\u0644\u0645\u0633\u0627\u0648\u0627\u0629. \u0648\u0647\u0648 1.} \\\\ \\\\\n\n(4) &amp; \\begin{array}{ll}\\phantom{-|}5&#039;7 \\div 4 &amp;= 1 \\\\ \\color{red}-|\\underline{4}\\color{black} &amp; \\\\ \\color{red}\\phantom{-|}1\\color{black} &amp;  \\end{array} \\\\ &amp; \\text{\u0636\u0631\u0628 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0641\u064a \u0627\u0644\u0645\u0642\u0633\u0648\u0645 \u0639\u0644\u064a\u0647 \u0648\u0637\u0631\u062d\u0647\u0627 \u0645\u0646 \u0627\u0644\u0642\u064a\u0645\u0629 \u0627\u0644\u0645\u062e\u062a\u0627\u0631\u0629.} \\\\ \\\\\n\n(5) &amp; \\begin{array}{ll}\\phantom{-|}5&#039;\\color{red}7&#039;\\color{black} \\div 4 &amp;= 1 \\\\ -|\\underline{4} &amp; \\\\ \\phantom{-|}1\\color{red}7\\color{black} &amp;  \\end{array} \\\\ &amp; \\text{\u0627\u062e\u062a\u064a\u0627\u0631 \u0648\u0625\u0646\u0632\u0627\u0644 \u0627\u0644\u0631\u0642\u0645 \u0627\u0644\u062a\u0627\u0644\u064a.} \\\\ \\\\\n\n(6) &amp; \\begin{array}{ll}\\phantom{-|}5&#039;7&#039; \\div 4 &amp;= 1\\color{red}4\\color{black} \\\\ -|\\underline{4} &amp; \\\\ \\phantom{-|}17 &amp; \\\\ \\color{red}-|\\underline{16}\\color{black} &amp; \\\\ \\color{red}\\phantom{-|}1\\color{black}  \\end{array} \\\\ &amp; \\text{\u062a\u0643\u0631\u0627\u0631 \u0627\u0644\u062a\u0633\u0644\u0633\u0644 \u0645\u0639 \u0627\u0644\u0639\u062f\u062f \u0627\u0644\u0623\u062e\u064a\u0631 \u0627\u0644\u0645\u062a\u062d\u0635\u0644 \u0639\u0644\u064a\u0647.}\n\n\\end{array}\n\n<\/span>\n<p>\u062a\u0646\u062a\u0647\u064a \u0627\u0644\u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0639\u0646\u062f\u0645\u0627 \u062a\u0646\u0641\u062f \u0627\u0644\u0623\u0631\u0642\u0627\u0645 \u0627\u0644\u0645\u062a\u0628\u0642\u064a\u0629 \u0644\u0640\u00ab\u0625\u0646\u0632\u0627\u0644\u0647\u0627\u00bb\u060c \u0648\u064a\u0643\u0648\u0646 \u0627\u0644\u0646\u0627\u062a\u062c \u0647\u0648 \u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=14<\/span><\/span> \u0648\u0627\u0644\u0628\u0627\u0642\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=1<\/span><\/span>. \u0648\u0639\u0644\u0649 \u0648\u062c\u0647 \u0627\u0644\u062e\u0635\u0648\u0635\u060c \u064a\u0643\u0648\u0646 \u0627\u0644\u0628\u0627\u0642\u064a \u062f\u0627\u0626\u0645\u0627\u064b \u0623\u0635\u063a\u0631 \u0645\u0646 \u0627\u0644\u0645\u0642\u0633\u0648\u0645 \u0639\u0644\u064a\u0647.<\/p>\n<\/li>\n<li>\n<p>\u0644\u0646\u0641\u062a\u0631\u0636 \u0623\u0646\u0646\u0627 \u0646\u0631\u064a\u062f \u062d\u0633\u0627\u0628 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">132\\div 5<\/span><\/span>. \u0644\u062a\u062d\u0642\u064a\u0642 \u0630\u0644\u0643 \u0633\u0646\u062c\u0631\u064a \u0627\u0644\u062a\u0633\u0644\u0633\u0644 \u0627\u0644\u0622\u062a\u064a \u0645\u0646 \u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u062d\u0633\u0627\u0628\u064a\u0629:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n(1) &amp;1&#039;32 \\div 5 = \\\\\n\n&amp; \\text{\u0641\u0635\u0644 \u0623\u0648\u0644 \u0628\u0627\u062f\u0626\u0629 \u0645\u0646 \u0627\u0644\u0645\u0642\u0633\u0648\u0645 \u0648\u0627\u0644\u0628\u062d\u062b \u0639\u0646 \u0639\u062f\u062f \u0637\u0628\u064a\u0639\u064a}\\\\\n\n&amp; \\text{\u0625\u0630\u0627 \u0636\u064f\u0631\u0628 \u0641\u064a 5 \u0643\u0627\u0646 \u0623\u0635\u063a\u0631 \u0645\u0646 \u0623\u0648 \u0645\u0633\u0627\u0648\u064a\u0627\u064b \u0644\u0647\u0627. \u0648\u0625\u0630\u0627 \u0644\u0645 \u064a\u0645\u0643\u0646\u060c}\\\\\n\n&amp; \\text{\u0646\u0636\u0645 \u0627\u0644\u0631\u0642\u0645 \u0627\u0644\u062a\u0627\u0644\u064a \u062d\u062a\u0649 \u064a\u0635\u0628\u062d \u0630\u0644\u0643 \u0645\u0645\u0643\u0646\u0627\u064b.} \\\\ \\\\\n\n(2) &amp;\\begin{array}{ll}\\phantom{-|}13&#039;2 \\div 5 &amp;= 2 \\\\ -|\\underline{10} &amp; \\\\ \\phantom{-|1}3 &amp; \\end{array}\\\\\n\n&amp; \\text{\u0639\u0646\u062f\u0645\u0627 \u062a\u0646\u062c\u062d \u0627\u0644\u062e\u0637\u0648\u0629 \u0627\u0644\u0633\u0627\u0628\u0642\u0629\u060c \u0646\u064f\u0646\u0641\u0651\u0630 \u0627\u0644\u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0628\u0635\u0648\u0631\u0629 \u0627\u0639\u062a\u064a\u0627\u062f\u064a\u0629.} \\\\ \\\\\n\n(3) &amp;\\begin{array}{ll}\\phantom{-|}13&#039;2&#039; \\div 5 &amp;= 26 \\\\ -|\\underline{10} &amp; \\\\ \\phantom{-|1}32 &amp;  \\\\ \\phantom{\\,} -|\\underline{30} &amp; \\\\ \\phantom{-|30} 2 &amp;  \\end{array}\n\n\\end{array}<\/span>\n<\/li>\n<li>\n<p>\u0625\u0630\u0627 \u0643\u0627\u0646 \u0644\u0644\u0645\u0642\u0633\u0648\u0645 \u0639\u062f\u062f \u0623\u0643\u0628\u0631 \u0645\u0646 \u0627\u0644\u0623\u0631\u0642\u0627\u0645\u060c \u064a\u064f\u0646\u0641\u0651\u064e\u0630 \u0627\u0644\u0625\u062c\u0631\u0627\u0621 \u0628\u0627\u0644\u0637\u0631\u064a\u0642\u0629 \u0646\u0641\u0633\u0647\u0627. \u0641\u0639\u0644\u0649 \u0633\u0628\u064a\u0644 \u0627\u0644\u0645\u062b\u0627\u0644\u060c \u0639\u0646\u062f \u062d\u0633\u0627\u0628 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">3521\\div 12<\/span><\/span> \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{ll}\n\n\\phantom{-|}35&#039;2&#039;1&#039; \\div 12 &amp; = 293 \\\\\n\n-|\\underline{24} &amp; \\\\\n\n\\phantom{-|}112 &amp; \\\\\n\n-|\\underline{108} &amp; \\\\\n\n\\phantom{-|10}41 &amp; \\\\\n\n\\phantom{0}-|\\underline{36} &amp; \\\\\n\n\\phantom{10-|}5 &amp; \\\\\n\n\\end{array}<\/span>\n<\/li>\n<li>\n<p>\u0625\u0630\u0627 \u0643\u0627\u0646 \u0623\u062d\u062f \u0627\u0644\u0639\u062f\u062f\u064a\u0646 \u0633\u0627\u0644\u0628\u0627\u064b\u060c \u0641\u0647\u0646\u0627\u0643 \u0637\u0631\u064a\u0642\u062a\u0627\u0646 \u0645\u0642\u0628\u0648\u0644\u062a\u0627\u0646 \u0644\u0639\u0631\u0636 \u0627\u0644\u0646\u062a\u064a\u062c\u0629. \u0648\u0645\u0639 \u0630\u0644\u0643\u060c \u0641\u0625\u0646 \u0627\u0644\u0635\u064a\u063a\u0629 \u0627\u0644\u0642\u064a\u0627\u0633\u064a\u0629 \u0641\u064a \u0646\u0638\u0631\u064a\u0629 \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0647\u064a \u062a\u0644\u0643 \u0627\u0644\u062a\u064a \u064a\u0643\u0648\u0646 \u0641\u064a\u0647\u0627 \u0627\u0644\u0628\u0627\u0642\u064a \u0645\u0648\u062c\u0628\u0627\u064b\u060c \u0648\u0645\u0646 \u062b\u0645 \u0645\u062a\u0633\u0642\u0629 \u0645\u0639 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629. \u0639\u0644\u0649 \u0633\u0628\u064a\u0644 \u0627\u0644\u0645\u062b\u0627\u0644\u060c \u0625\u0630\u0627 \u062d\u0633\u0628\u0646\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598\\div 21<\/span><\/span>\u060c \u0641\u0646\u062d\u0635\u0644 \u0639\u0644\u0649 \u0627\u0644\u062a\u0637\u0648\u064a\u0631\u0627\u062a \u0627\u0644\u0645\u0645\u0643\u0646\u0629 \u0627\u0644\u0622\u062a\u064a\u0629\u060c \u0648\u0643\u0644\u0651\u064c \u0645\u0646\u0647\u0627 \u0635\u062d\u064a\u062d \u0628\u0648\u0635\u0641\u0647 \u0645\u062a\u0637\u0627\u0628\u0642\u0629:<\/p>\n<ul>\n<li>\n        <b>\u0645\u0639 \u0628\u0627\u0642\u064d \u0633\u0627\u0644\u0628:<\/b> \u064a\u062a\u0645 \u0627\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u064a\u0647 \u0628\u0625\u0632\u0627\u0644\u0629 \u0627\u0644\u0625\u0634\u0627\u0631\u0629 \u0641\u064a \u0628\u062f\u0627\u064a\u0629 \u0627\u0644\u0639\u0645\u0644\u064a\u0629 \u062b\u0645 \u0625\u0639\u0627\u062f\u062a\u0647\u0627 \u0641\u064a \u0627\u0644\u0646\u0647\u0627\u064a\u0629.<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{ll}\n\n\\phantom{-}59&#039;8&#039; \\div 21 &amp;= 28 \\\\\n\n-|\\underline{42} &amp; \\\\\n\n\\phantom{-|&#039;}178 &amp; \\\\\n\n\\phantom{\\,}-|\\underline{168} &amp; \\\\\n\n\\phantom{-|00}10 &amp; \\\\\n\n\\end{array}<\/span>\n<p>\u0648\u0645\u0646 \u0647\u0630\u0627 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">598 = 21 \\times 28 + 10<\/span>\n<p>\u062b\u0645 \u0625\u0630\u0627 \u0636\u0631\u0628\u0646\u0627 \u0637\u0631\u0641\u064a \u0627\u0644\u0645\u0633\u0627\u0648\u0627\u0629 \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-1<\/span><\/span> \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598 = 21 \\times (-28) - 10<\/span>\n<p>\u0648\u0639\u0644\u064a\u0647\u060c \u0641\u064a \u0647\u0630\u0627 \u0627\u0644\u062a\u0645\u062b\u064a\u0644 \u064a\u0643\u0648\u0646 \u0646\u0627\u062a\u062c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598\\div 21<\/span><\/span> \u0647\u0648 \u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=-28<\/span><\/span> \u0648\u0627\u0644\u0628\u0627\u0642\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=-10<\/span><\/span>.<\/p>\n<p>\u0648\u0625\u0630\u0627 \u0636\u0631\u0628\u0646\u0627 \u0645\u0631\u0629 \u0623\u062e\u0631\u0649 \u0637\u0631\u0641\u064a \u0627\u0644\u0645\u0633\u0627\u0648\u0627\u0629 \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-1<\/span><\/span> \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">598 = (-21)\\times(-28) + 10<\/span>\n<p>\u0648\u0628\u0630\u0644\u0643 \u064a\u0643\u0648\u0646 \u0646\u0627\u062a\u062c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">598\\div (-21)<\/span><\/span> \u0647\u0648 \u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=-28<\/span><\/span> \u0648\u0627\u0644\u0628\u0627\u0642\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=10<\/span><\/span>.<\/p>\n<\/li>\n<li>\n        <b>\u0645\u0639 \u0628\u0627\u0642\u064d \u0645\u0648\u062c\u0628 \u0648\u0645\u062a\u0633\u0642 \u0645\u0639 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629:<\/b> \u0627\u0646\u0637\u0644\u0627\u0642\u0627\u064b \u0645\u0646 \u0627\u0644\u062a\u0637\u0648\u064a\u0631 \u0627\u0644\u0633\u0627\u0628\u0642 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598 = 21 \\times (-28) - 10<\/span>\n<p>\u062b\u0645 \u0628\u0625\u0636\u0627\u0641\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=21-21<\/span><\/span> \u0625\u0644\u0649 \u0627\u0644\u0637\u0631\u0641 \u0627\u0644\u0623\u064a\u0645\u0646 \u0645\u0646 \u0647\u0630\u0647 \u0627\u0644\u0645\u0633\u0627\u0648\u0627\u0629 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598 = 21 \\times (-28) - 10 \\color{red}+ 21 - 21\\color{black} = 21\\times(-29) + 11<\/span>\n<p>\u0648\u0639\u0644\u064a\u0647\u060c \u0648\u0628\u0627\u0644\u0627\u0644\u062a\u0632\u0627\u0645 \u0628\u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629\u060c \u0641\u0625\u0646 \u0646\u062a\u064a\u062c\u0629 \u062d\u0633\u0627\u0628 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598\\div 21<\/span><\/span> \u0647\u064a \u062e\u0627\u0631\u062c \u0642\u0633\u0645\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=-29<\/span><\/span> \u0648\u0628\u0627\u0642\u064d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=11<\/span><\/span>.<\/p>\n<\/li>\n<\/ul>\n<p>\u0641\u064a\u0645\u0627 \u064a\u062a\u0639\u0644\u0651\u0642 \u0628\u0645\u0627 \u0647\u0648 \u0627\u0635\u0637\u0644\u0627\u062d\u064a \u0641\u064a \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629\u060c \u0641\u0625\u0646 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0630\u0627\u062a \u0627\u0644\u0628\u0627\u0642\u064a \u0627\u0644\u0645\u0648\u062c\u0628 \u0641\u0642\u0637 \u0647\u064a \u0627\u0644\u062a\u064a \u062a\u0645\u062b\u0651\u0644 \u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629 \u0648\u0627\u0644\u0628\u0627\u0642\u064a \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u064a\u0646. \u0648\u0645\u0639 \u0630\u0644\u0643\u060c \u0641\u0625\u0646 \u0643\u0644\u0627 \u0627\u0644\u062a\u0639\u0628\u064a\u0631\u064a\u0646 \u0647\u0648\u064a\u062a\u0627\u0646 \u0635\u062d\u064a\u062d\u062a\u0627\u0646 \u0648\u064a\u0645\u0643\u0646 \u0623\u0646 \u064a\u0643\u0648\u0646\u0627 \u0645\u0641\u064a\u062f\u064a\u0646 \u0641\u064a \u0633\u064a\u0627\u0642\u0627\u062a \u0645\u062e\u062a\u0644\u0641\u0629. \u0648\u0639\u0644\u0649 \u0627\u0644\u0631\u063a\u0645 \u0645\u0646 \u0623\u0646 \u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0637\u0648\u064a\u0644\u0629 \u0642\u062f \u062a\u064f\u0646\u062a\u062c \u0628\u0648\u0627\u0642\u064a \u0633\u0627\u0644\u0628\u0629\u060c \u0641\u0625\u0646 \u0630\u0644\u0643 \u064a\u0643\u0648\u0646 \u0639\u0645\u0644\u064a\u0627\u064b \u0644\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 \u0627\u0644\u0646\u062a\u0627\u0626\u062c \u0628\u0637\u0631\u064a\u0642\u0629 \u0622\u0644\u064a\u0629 \u0648\u0625\u062c\u0631\u0627\u0621 \u0645\u0639\u0627\u0644\u062c\u0627\u062a \u062c\u0628\u0631\u064a\u0629 \u0645\u0646 \u062f\u0648\u0646 \u062e\u0637\u0648\u0627\u062a \u0625\u0636\u0627\u0641\u064a\u0629. \u0648\u0645\u0646 \u062c\u0647\u0629 \u0623\u062e\u0631\u0649\u060c \u0641\u0625\u0646 \u0627\u0644\u0646\u062a\u0627\u0626\u062c \u0630\u0627\u062a \u0627\u0644\u0628\u0627\u0642\u064a \u0627\u0644\u0645\u0648\u062c\u0628\u060c \u0627\u0644\u0645\u062a\u0633\u0642\u0629 \u0645\u0639 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629\u060c \u062a\u0633\u0645\u062d \u0628\u062a\u062b\u0628\u064a\u062a \u0645\u0645\u062b\u0651\u0644 \u0642\u064a\u0627\u0633\u064a \u0644\u0643\u0644 \u0641\u0626\u0629 \u0628\u0648\u0627\u0642\u064a (\u0639\u0644\u0649 \u0633\u0628\u064a\u0644 \u0627\u0644\u0645\u062b\u0627\u0644\u060c \u0636\u0645\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{0,1,\\dots,n-1\\}<\/span><\/span>)\u060c \u0645\u0645\u0627 \u064a\u0633\u0647\u0651\u0644 \u0648\u0633\u0645 \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0635\u062d\u064a\u062d\u0629 \u0628\u062f\u0642\u0629 \u0648\u0645\u0646 \u062f\u0648\u0646 \u0644\u0628\u0633. \u0648\u064a\u062c\u062f\u0631 \u0627\u0644\u062a\u0646\u0628\u064a\u0647 \u0625\u0644\u0649 \u0623\u0646 \u0647\u0646\u0627\u0643 \u0623\u064a\u0636\u0627\u064b \u0627\u0635\u0637\u0644\u0627\u062d\u0627\u062a \u0623\u062e\u0631\u0649 \u0644\u0644\u0645\u0645\u062b\u0651\u0644\u0627\u062a\u060c \u0645\u062b\u0644 \u0627\u0644\u0628\u0648\u0627\u0642\u064a \u0627\u0644\u0645\u062a\u0645\u0627\u062b\u0644\u0629\u060c \u0648\u0647\u064a \u0635\u0627\u0644\u062d\u0629 \u0628\u0627\u0644\u0642\u062f\u0631 \u0646\u0641\u0633\u0647 \u0645\u062a\u0649 \u062a\u0645 \u062a\u062b\u0628\u064a\u062a \u0627\u0644\u0642\u0627\u0639\u062f\u0629.<\/p>\n<\/li>\n<\/ol>\n<p><a name=\"6\"><\/a><\/p>\n<h2>\u0627\u0644\u062e\u0644\u0627\u0635\u0629<\/h2>\n<p>\u062a\u0636\u0645\u0646 \u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629 \u0623\u0646 \u0643\u0644 \u0642\u0633\u0645\u0629 \u0635\u062d\u064a\u062d\u0629 \u064a\u0645\u0643\u0646 \u0627\u0644\u062a\u0639\u0628\u064a\u0631 \u0639\u0646\u0647\u0627 \u0628\u0635\u0648\u0631\u0629 \u0648\u062d\u064a\u062f\u0629 \u0639\u0644\u0649 \u0627\u0644\u0634\u0643\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span> \u0645\u0639 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>\u060c \u0648\u0647\u0648 \u0645\u0627 \u064a\u062d\u062f\u0651\u062f \u0645\u0639\u064a\u0627\u0631\u0627\u064b \u0642\u064a\u0627\u0633\u064a\u0627\u064b \u0644\u062a\u0641\u0633\u064a\u0631 \u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629 \u0648\u0627\u0644\u0628\u0627\u0642\u064a \u062d\u062a\u0649 \u0639\u0646\u062f \u062f\u062e\u0648\u0644 \u0623\u0639\u062f\u0627\u062f \u0633\u0627\u0644\u0628\u0629. \u0648\u0644\u0627 \u062a\u0639\u062f\u0651 \u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0637\u0648\u064a\u0644\u0629 \u0633\u0648\u0649 \u062a\u0637\u0628\u064a\u0642 \u0639\u0645\u0644\u064a \u0644\u0647\u0630\u0627 \u0627\u0644\u062a\u0641\u0643\u064a\u0643\u060c \u0642\u0627\u0626\u0645 \u0639\u0644\u0649 \u0627\u0644\u062a\u0645\u062b\u064a\u0644 \u0627\u0644\u0645\u0648\u0636\u0639\u064a\u060c \u0648\u062a\u062a\u064a\u062d \u062d\u0633\u0627\u0628 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> \u0628\u0637\u0631\u064a\u0642\u0629 \u0622\u0644\u064a\u0629. \u0648\u0623\u062e\u064a\u0631\u0627\u064b\u060c \u0641\u0625\u0646 \u0647\u0630\u0647 \u0627\u0644\u0635\u064a\u063a\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629 \u0644\u0627 \u062a\u0642\u062a\u0635\u0631 \u0639\u0644\u0649 \u062a\u062c\u0646\u0651\u0628 \u0627\u0644\u0627\u0644\u062a\u0628\u0627\u0633 \u0641\u062d\u0633\u0628\u060c \u0628\u0644 \u062a\u0631\u062a\u0628\u0637 \u0623\u064a\u0636\u0627\u064b \u0628\u0635\u0648\u0631\u0629 \u0637\u0628\u064a\u0639\u064a\u0629 \u0628\u0627\u0644\u062d\u0633\u0627\u0628\u064a\u0627\u062a \u0627\u0644\u0645\u0639\u064a\u0627\u0631\u064a\u0629\u061b \u0625\u0630 \u064a\u0639\u0645\u0644 \u0627\u0644\u0628\u0627\u0642\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> \u0645\u0645\u062b\u0651\u0644\u0627\u064b \u0642\u064a\u0627\u0633\u064a\u0627\u064b \u0644\u0641\u0626\u0629 \u0627\u0644\u0628\u0648\u0627\u0642\u064a \u0644\u0644\u0639\u062f\u062f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u0628\u062a\u0631\u062f\u064a\u062f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>\u060c \u0648\u0647\u0648 \u0627\u0644\u0623\u0633\u0627\u0633 \u0644\u0644\u0639\u062f\u064a\u062f \u0645\u0646 \u0627\u0644\u062a\u0642\u0646\u064a\u0627\u062a \u0641\u064a \u0646\u0638\u0631\u064a\u0629 \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0648\u0641\u064a \u0627\u0644\u062a\u0637\u0628\u064a\u0642\u0627\u062a \u0627\u0644\u062d\u0627\u0633\u0648\u0628\u064a\u0629 \u0627\u0644\u062a\u064a \u0633\u064a\u062a\u0645 \u062a\u0646\u0627\u0648\u0644\u0647\u0627 \u0641\u064a \u0639\u0631\u0648\u0636 \u0644\u0627\u062d\u0642\u0629.<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h2>\u062a\u0645\u0627\u0631\u064a\u0646 \u0645\u0642\u062a\u0631\u062d\u0629 \u0648\u0645\u062d\u0644\u0648\u0644\u0629<\/h2>\n<h3>\u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629<\/h3>\n<ol>\n<li><strong>(\u0645\u062d\u0644\u0648\u0644)<\/strong> \u0623\u0648\u062c\u062f <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> \u0648<span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> \u0628\u062d\u064a\u062b\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">137 = 9q + r,\\qquad 0\\le r\\lt 9.<\/span>\n<p><strong>\u0627\u0644\u062d\u0644:<\/strong> \u0627\u0628\u062d\u062b \u0639\u0646 \u0623\u0643\u0628\u0631 \u0639\u062f\u062f \u0625\u0630\u0627 \u0636\u064f\u0631\u0628 \u0641\u064a <span class=\"katex-eq\" data-katex-display=\"false\">9<\/span> \u0644\u0627 \u064a\u062a\u062c\u0627\u0648\u0632 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">137<\/span><\/span>. \u0644\u062f\u064a\u0646\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">15\\times 9 = 135<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">16\\times 9 = 144<\/span><\/span>\u060c \u0648\u0645\u0646 \u062b\u0645 \u0641\u0625\u0646 \u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0645\u0637\u0644\u0648\u0628 \u0647\u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=15<\/span><\/span>. \u0627\u0644\u0622\u0646 \u0627\u062d\u0633\u0628 \u0627\u0644\u0628\u0627\u0642\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=137-9q<\/span><\/span> \u0648\u062a\u062d\u0642\u0651\u0642 \u0645\u0646 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r\\lt 9<\/span><\/span>. \u0648\u0628\u0625\u062c\u0631\u0627\u0621 \u0627\u0644\u062a\u062d\u0642\u0651\u0642 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nr=137-9\\cdot 15=137-135=2.\n\n<\/span>\n<p>\u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a\u060c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">137=9\\cdot 15+2<\/span><\/span> \u0645\u0639 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le 2\\lt 9<\/span><\/span>.<\/p>\n<\/li>\n<li><strong>(\u0645\u062d\u0644\u0648\u0644)<\/strong> \u0623\u0648\u062c\u062f <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> \u0648<span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> \u0628\u062d\u064a\u062b\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2025 = 37q + r,\\qquad 0\\le r\\lt 37.<\/span>\n<p><strong>\u0627\u0644\u062d\u0644:<\/strong> \u0627\u0628\u062d\u062b \u0639\u0646 \u0623\u0643\u0628\u0631 \u0639\u062f\u062f \u0625\u0630\u0627 \u0636\u064f\u0631\u0628 \u0641\u064a <span class=\"katex-eq\" data-katex-display=\"false\">37<\/span> \u0644\u0627 \u064a\u062a\u062c\u0627\u0648\u0632 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2025<\/span><\/span>. \u0644\u0627\u062d\u0638 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">37\\times 54 = 1998<\/span><\/span> \u0648<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">37\\times 55 = 2035<\/span><\/span>\u060c \u0648\u0645\u0646 \u062b\u0645 \u0641\u0625\u0646 \u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0645\u0637\u0644\u0648\u0628 \u0647\u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=54<\/span><\/span>. \u0627\u0644\u0622\u0646 \u0627\u062d\u0633\u0628 \u0627\u0644\u0628\u0627\u0642\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=2025-37q<\/span><\/span> \u0648\u062a\u062d\u0642\u0651\u0642 \u0645\u0646 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r\\lt 37<\/span><\/span>. \u0648\u0628\u0625\u062c\u0631\u0627\u0621 \u0627\u0644\u062a\u062d\u0642\u0651\u0642 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nr=2025-37\\cdot 54=2025-1998=27.\n\n<\/span>\n<p>\u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a\u060c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2025=37\\cdot 54+27<\/span><\/span> \u0645\u0639 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le 27\\lt 37<\/span><\/span>.<\/p>\n<\/li>\n<li>\u0623\u0648\u062c\u062f <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> \u0648<span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> \u0628\u062d\u064a\u062b\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">745 = 23q + r,\\qquad 0\\le r\\lt 23.<\/span>\n<\/li>\n<li>\u0623\u0648\u062c\u062f <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> \u0648<span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> \u0628\u062d\u064a\u062b\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-314 = 11q + r,\\qquad 0\\le r\\lt 11.<\/span>\n<\/li>\n<li>\u0623\u0648\u062c\u062f <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> \u0648<span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> \u0628\u062d\u064a\u062b\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">598 = (-21)q + r,\\qquad 0\\le r\\lt |-21|.<\/span>\n<\/li>\n<li><strong>\u0628\u0627\u0642\u064d \u0645\u064f\u062b\u0628\u0651\u062a \u0648\u0627\u0644\u0628\u062d\u062b \u0639\u0646 \u0623\u0639\u062f\u0627\u062f \u0635\u062d\u064a\u062d\u0629.<\/strong> \u0644\u064a\u0643\u0646 <span class=\"katex-eq\" data-katex-display=\"false\">a=12<\/span>.\n<p>(\u0623) \u0635\u0650\u0641 \u0645\u062c\u0645\u0648\u0639\u0629 \u062c\u0645\u064a\u0639 \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0635\u062d\u064a\u062d\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> \u0627\u0644\u062a\u064a \u064a\u0643\u0648\u0646 \u0628\u0627\u0642\u0650\u064a \u0642\u0633\u0645\u062a\u0647\u0627 \u0627\u0644\u0625\u0642\u0644\u064a\u062f\u064a\u0629 \u0639\u0644\u0649 <span class=\"katex-eq\" data-katex-display=\"false\">12<\/span> \u0647\u0648 <span class=\"katex-eq\" data-katex-display=\"false\">r=5<\/span>.<\/p>\n<p>(\u0628) \u0623\u0648\u062c\u062f \u0623\u0635\u063a\u0631 \u0639\u062f\u062f \u0635\u062d\u064a\u062d <span class=\"katex-eq\" data-katex-display=\"false\">b\\gt 1000<\/span> \u064a\u062d\u0642\u0642 \u0645\u0627 \u0633\u0628\u0642\u060c \u0648\u062d\u062f\u0651\u062f \u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> \u0627\u0644\u0645\u0648\u0627\u0641\u0642.<\/p>\n<\/li>\n<\/ol>\n<h3>\u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0637\u0648\u064a\u0644\u0629<\/h3>\n<p>\u0641\u064a \u0643\u0644 \u062d\u0627\u0644\u0629\u060c \u0637\u0628\u0651\u0642 \u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629 \u0627\u0644\u0637\u0648\u064a\u0644\u0629 \u0644\u062d\u0633\u0627\u0628 \u062e\u0627\u0631\u062c \u0627\u0644\u0642\u0633\u0645\u0629 <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> \u0648\u0627\u0644\u0628\u0627\u0642\u064a <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span>. \u0648\u062d\u0648\u0651\u0644 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0625\u0644\u0649 \u0642\u0633\u0645\u0629 \u0625\u0642\u0644\u064a\u062f\u064a\u0629 \u0639\u0646\u062f\u0645\u0627 \u062a\u0639\u0637\u064a \u0627\u0644\u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0628\u0627\u0642\u064a\u0627\u064b \u0633\u0627\u0644\u0628\u0627\u064b.\n<\/p>\n<ol>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">84\\div 6.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">197\\div 8.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1256\\div 7.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-3521\\div 12.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-98765\\div 24.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">845\\div -13.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-12345\\div -37.<\/span><\/span><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629 \u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u062d\u0635\u0629 \u0633\u0646\u0637\u0648\u0651\u0631 \u062e\u0648\u0627\u0631\u0632\u0645\u064a\u0629 \u0627\u0644\u0642\u0633\u0645\u0629 \u0628\u0648\u0635\u0641\u0647\u0627 \u0627\u0644\u0645\u0628\u062f\u0623 \u0627\u0644\u0630\u064a \u064a\u0636\u0641\u064a \u0627\u0644\u0637\u0627\u0628\u0639 \u0627\u0644\u0631\u0633\u0645\u064a\u060c \u0639\u0644\u0649 \u0645\u0633\u062a\u0648\u0649 \u0627\u0644\u0623\u0639\u062f\u0627\u062f 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Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.","sameAs":["http:\/\/toposuranos.com\/material"],"url":"https:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/"}]}},"_links":{"self":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/35994","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/comments?post=35994"}],"version-history":[{"count":0,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/35994\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media\/35980"}],"wp:attachment":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media?parent=35994"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=35994"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=35994"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}