{"id":35665,"date":"2025-12-26T18:56:31","date_gmt":"2025-12-26T18:56:31","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35665"},"modified":"2025-12-26T18:56:31","modified_gmt":"2025-12-26T18:56:31","slug":"%e4%bb%80%e4%b9%88%e6%98%af%e5%8f%af%e6%95%b4%e9%99%a4%e6%80%a7%ef%bc%9f","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/zh\/%e4%bb%80%e4%b9%88%e6%98%af%e5%8f%af%e6%95%b4%e9%99%a4%e6%80%a7%ef%bc%9f\/","title":{"rendered":"\u4ec0\u4e48\u662f\u53ef\u6574\u9664\u6027\uff1f"},"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>\u53ef\u9664\u6027<\/h1>\n<p style=\"text-align:center;\"><em><br \/>\n<strong>\u53ef\u9664\u6027<\/strong>\u662f\u6570\u8bba\u771f\u6b63\u7684\u8d77\u70b9\uff0c\u56e0\u4e3a\u5b83\u5c06\u6574\u6570\u8f6c\u5316\u4e3a\u4e00\u4e2a\u5177\u6709\u7ed3\u6784\u7684\u7cfb\u7edf\uff1a\u4f60\u4e0d\u518d\u5c06\u6570\u5b57\u4ec5\u4ec5\u89c6\u4e3a\u201c\u6570\u91cf\u201d\uff0c\u800c\u662f\u5c06\u5176\u770b\u4f5c\u5f7c\u6b64\u4e4b\u95f4\u53ef\u4ee5\u6216\u4e0d\u53ef\u4ee5\u76f8\u4e92\u5951\u5408\u7684\u7ec4\u6210\u90e8\u5206\u3002\u501f\u52a9\u5355\u4e00\u7684\u7ed3\u6784 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>\uff0c\u4f60\u53ef\u4ee5\u8868\u8fbe\u4ece\u5316\u7b80\u4e0e\u56e0\u5f0f\u5206\u89e3\u7684\u5224\u5b9a\u51c6\u5219\uff0c\u5230\u8bf8\u5982\u6b27\u51e0\u91cc\u5f97\u7b97\u6cd5\u7b49\u6838\u5fc3\u8fc7\u7a0b\u7684\u672c\u8d28\uff0c\u800c\u8be5\u7b97\u6cd5\u4f7f\u5f97\u5373\u4f7f\u9762\u5bf9\u5927\u6574\u6570\uff0c\u4e5f\u80fd\u591f\u5728\u6781\u77ed\u65f6\u95f4\u5185\u8ba1\u7b97\u6700\u5927\u516c\u7ea6\u6570\u3002\u6b64\u5916\uff0c\u53ef\u9664\u6027\u8fd8\u662f\u5728\u5e94\u7528\u6570\u5b66\u4e0e\u8ba1\u7b97\u673a\u79d1\u5b66\u4e2d\u53cd\u590d\u51fa\u73b0\u7684\u601d\u60f3\u7684\u6280\u672f\u57fa\u7840\uff1a\u540c\u4f59\u3001\u6a21\u7b97\u672f\u3001\u9a8c\u8bc1\u673a\u5236\u3001\u7f16\u7801\uff0c\u4ee5\u53ca\uff08\u5728\u540e\u7eed\u5185\u5bb9\u4e2d\uff09\u5bc6\u7801\u5b66\u3002\u4ece\u672c\u8d28\u4e0a\u8bf4\uff0c\u638c\u63e1\u53ef\u9664\u6027\uff0c\u5c31\u662f\u5b66\u4f1a\u5728\u6574\u6570\u4e2d\u8bc6\u522b\u90a3\u4e9b\u4e0d\u53ef\u89c1\u7684\u6a21\u5f0f\uff0c\u5e76\u5c06\u5176\u8f6c\u5316\u4e3a\u59cb\u7ec8\u6709\u6548\u7684\u64cd\u4f5c\u8fc7\u7a0b\u3002<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><b>\u5b66\u4e60\u76ee\u6807<\/b><br \/>\n\u5b8c\u6210\u672c\u8bb2\u4e49\u540e\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a\n<\/p>\n<ol>\n<li><strong>\u7406\u89e3<\/strong>\u6574\u6570\u4e4b\u95f4\u7684\u53ef\u9664\u5173\u7cfb\u3002<\/li>\n<li><strong>\u7406\u89e3<\/strong>\u53ef\u9664\u6027\u7684\u5b9a\u4e49\u53ca\u5176\u6027\u8d28\u3002<\/li>\n<li><strong>\u53d1\u5c55<\/strong>\u4e0e\u53ef\u9664\u6027\u76f8\u5173\u7684\u6570\u5b66\u7ed3\u8bba\u548c\u5b9a\u7406\u7684\u8bc1\u660e\u80fd\u529b\u3002<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><b><u>\u5185\u5bb9\u7d22\u5f15<\/u><\/b><br \/>\n<a href=\"#1\">\u53ef\u9664\u6027\u7684\u5b9a\u4e49<\/a><br \/>\n<a href=\"#2\">\u53ef\u9664\u6027\u7684\u57fa\u672c\u6027\u8d28<\/a><br \/>\n<a href=\"#3\">\u4e60\u9898<\/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>\u53ef\u9664\u6027\u7684\u5b9a\u4e49<\/h2>\n<p style=\"text-align: justify;\">\n\u201c<em>a \u6574\u9664 b<\/em>\u201d\u8fd9\u4e00\u975e\u6b63\u5f0f\u60f3\u6cd5\uff0c\u5728\u5c06\u5176\u8868\u8ff0\u4e3a\u6574\u6570\u4e4b\u95f4\u7684\u5173\u7cfb\u65f6\u624d\u53d8\u5f97\u7cbe\u786e\u3002\u6211\u4eec\u79f0\u4e00\u4e2a\u6574\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> <strong>\u6574\u9664<\/strong> \u4e00\u4e2a\u6574\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span>\uff0c\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u53ef\u4ee5\u8868\u793a\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u7684\u4e00\u4e2a\u7cbe\u786e\u500d\u6570\u3002\u8be5\u5b9a\u4e49\u662f\u672c\u8bb2\u4e49\u5176\u4f59\u5185\u5bb9\u7684\u57fa\u7840\uff0c\u56e0\u4e3a\u5b83\u5c06\u8bf8\u5982\u201c\u6070\u597d\u6574\u9664\u201d\u4e4b\u7c7b\u7684\u8868\u8ff0\u8f6c\u5316\u4e3a\u4e00\u4e2a\u53ef\u9a8c\u8bc1\u7684\u5224\u5b9a\u6807\u51c6\u3002\n<\/p>\n<p style=\"text-align: justify;\">\n<strong>\u5b9a\u4e49\u3002<\/strong> \u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>\u3002\u6211\u4eec\u79f0 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u6574\u9664 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span>\uff0c\u5e76\u8bb0\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>\uff0c\u5f53\u4e14\u4ec5\u5f53\u5b58\u5728\u4e00\u4e2a\u6574\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> \u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=ka<\/span><\/span>\u3002\u5426\u5219\uff0c\u8bb0\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\nmid b<\/span><\/span>\u3002\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\u5728\u8be5\u5b9a\u4e49\u4e2d\uff0c\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> \u88ab\u79f0\u4e3a\u4e0e\u8be5\u53ef\u9664\u6027\u76f8\u5173\u7684 <strong>\u5546<\/strong>\uff08\u6216\u56e0\u5b50\uff09\u3002\u4f8b\u5982\uff0c\u65ad\u8a00 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">6\\mid 42<\/span><\/span> \u7b49\u4ef7\u4e8e\u65ad\u8a00\u5b58\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> \u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">42=6k<\/span><\/span>\uff1b\u5728\u6b64\u60c5\u5f62\u4e0b\uff0c\u53ea\u9700\u53d6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=7<\/span><\/span>\u3002\n<\/p>\n<h3>\u9700\u8981\u6ce8\u610f\u7684\u91cd\u8981\u4e8b\u9879<\/h3>\n<ul>\n<li>\n    \u6761\u4ef6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span> \u662f\u81f3\u5173\u91cd\u8981\u7684\u3002\u56e0\u4e3a\u5982\u679c\u8bd5\u56fe\u5141\u8bb8 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=0<\/span><\/span>\uff0c\u5219\u53ef\u9664\u6027\u7684\u6761\u4ef6\u5c06\u8981\u6c42\u5b58\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> \u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0\\cdot k<\/span><\/span>\u3002\u7136\u800c\uff0c\u5bf9\u6240\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> \u90fd\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\cdot k=0<\/span><\/span>\uff0c\u56e0\u6b64\u552f\u4e00\u7684\u53ef\u80fd\u6027\u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span>\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u5e76\u4e0d\u5b58\u5728\u4e00\u4e2a\u7531\u8be5\u5173\u7cfb\u201c\u786e\u5b9a\u201d\u7684 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span>\uff0c\u56e0\u4e3a\u4efb\u610f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> \u90fd\u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=0\\cdot k<\/span><\/span>\u3002\u6362\u8a00\u4e4b\uff0c\u975e\u6b63\u5f0f\u7684\u8868\u8fbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=b\/a<\/span><\/span> \u5c06\u53d8\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0\/0<\/span><\/span>\uff0c\u800c\u8fd9\u662f\u6ca1\u6709\u5b9a\u4e49\u7684\u3002\u4e3a\u4e86\u907f\u514d\u8fd9\u79cd\u9000\u5316\u60c5\u5f62\uff08\u5728\u5176\u4e2d\u201c\u5546\u201d\u7684\u6982\u5ff5\u4e0d\u518d\u5177\u6709\u610f\u4e49\uff09\uff0c\u5fc5\u987b\u8981\u6c42 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>\u3002\u56e0\u6b64\uff0c\u5173\u7cfb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\mid b<\/span><\/span> \u4e0d\u88ab\u89c6\u4e3a\u6709\u6548\u3002\n<\/li>\n<li>\n        \u76f8\u53cd\u5730\uff0c\u5bf9\u4e8e\u6240\u6709\u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span> \u7684 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{Z}<\/span><\/span>\uff0c\u547d\u9898 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid 0<\/span><\/span> \u90fd\u6210\u7acb\uff0c\u56e0\u4e3a\u53ea\u9700\u53d6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span>\uff0c\u4fbf\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=a\\cdot 0<\/span><\/span>\u3002\n    <\/li>\n<\/ul>\n<p style=\"text-align: justify;\">\n\u7531\u8be5\u5b9a\u4e49\u53ef\u4ee5\u63a8\u51fa\u4e00\u4e2a\u6211\u4eec\u5c06\u53cd\u590d\u4f7f\u7528\u7684\u7b49\u4ef7\u8868\u8ff0\uff1a\u65ad\u8a00 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> \u7b49\u4ef7\u4e8e\u65ad\u8a00 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u5c5e\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u7684\u6574\u6570\u500d\u96c6\u5408\uff0c\u5373 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\in a\\mathbb{Z}<\/span><\/span>\uff0c\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mathbb{Z}=\\{ak:\\,k\\in\\mathbb{Z}\\}<\/span><\/span>\u3002\u8fd9\u79cd\u5199\u6cd5\u5f3a\u8c03\u4e86\u53ef\u9664\u6027\u5e76\u975e\u4e00\u79cd\u201c\u6280\u5de7\u201d\uff0c\u800c\u662f\u4e00\u79cd\u7528\u4e8e\u63cf\u8ff0 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{Z}<\/span><\/span> \u4e2d\u9ad8\u5ea6\u7ed3\u6784\u5316\u5b50\u96c6\u7684\u65b9\u5f0f\u3002\n<\/p>\n<p><a name=\"2\"><\/a><\/br><\/p>\n<h2>\u53ef\u9664\u6027\u7684\u57fa\u672c\u6027\u8d28<\/h2>\n<ul>\n<li><strong>\u81ea\u53cd\u6027\uff1a<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid a<\/span><\/span>\u3002<br \/>\n<u>\u8bc1\u660e<\/u>\uff1a<\/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{; $\\mathbb{Z}$ \u4e2d\u7684\u4e58\u6cd5\u5355\u4f4d\u5143}\\\\\n\n(2)&amp;\\vdash(\\exists k \\in \\mathbb{Z})(a=ka) &amp;\\text{; \u5b58\u5728\u91cf\u8bcd\u5f15\u5165 (1)}\\\\\n\n(3) &amp;\\vdash a \\mid a &amp;\\text{; \u53ef\u9664\u6027\u7684\u5b9a\u4e49 (2)} \\\\\n\n&amp;\\blacksquare &amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<li><strong>\u4f20\u9012\u6027\uff1a<\/strong> \u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\mid c<\/span><\/span>\uff0c\u5219 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>\u3002\n<p><u>\u8bc1\u660e<\/u>\uff1a<\/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{; \u53ef\u9664\u6027\u7684\u5b9a\u4e49\uff0c\u524d\u63d0}\\\\\n\n(2)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_2\\in\\mathbb{Z})(c=k_2b)  &amp;\\text{; \u53ef\u9664\u6027\u7684\u5b9a\u4e49\uff0c\u524d\u63d0}\\\\\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{; $\\exists$-\u5408\u5e76 (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{; \u7531(3)}\\\\\n\n(5)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_1,k_2\\in\\mathbb{Z})( c=k_1k_2a)  &amp;\\text{; \u7531(4)}\\\\\n\n&amp;\\text{\u91cf\u8bcd\u5185\u90e8\u7684\u4ee3\u6570\u8fd0\u7b97}&amp; \\\\\n\n(6)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k\\in\\mathbb{Z})( c=ka)  &amp;\\text{; \u7531(5)}\\\\\n\n&amp;\\text{$\\mathbb{Z}$ \u5bf9\u4e58\u6cd5\u7684\u5c01\u95ed\u6027}&amp; \\\\\n\n(7)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash a\\mid c  &amp;\\text{; \u53ef\u9664\u6027\u7684\u5b9a\u4e49 (6)}\\\\\n\n(8)&amp; \\vdash (a\\mid b \\wedge  b\\mid c) \\rightarrow  a\\mid c  &amp;\\text{; $\\wedge$-\u6d88\u53bb (7)}\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<li><strong>\u4e0e\u52a0\u6cd5\u548c\u51cf\u6cd5\u7684\u76f8\u5bb9\u6027\uff1a<\/strong> \u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>\uff0c\u5219 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (b+c)<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (b-c)<\/span><\/span>\u3002<br \/>\n<u>\u8bc1\u660e<\/u>\uff1a<\/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{; \u53ef\u9664\u6027\u7684\u5b9a\u4e49\uff0c\u524d\u63d0}\\\\\n\n(2)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_2 \\in \\mathbb{Z})(c=k_2 a) &amp;\\text{; \u53ef\u9664\u6027\u7684\u5b9a\u4e49\uff0c\u524d\u63d0}\\\\\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{; $\\exists$-\u5408\u5e76 (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{; \u7531(3)}\\\\\n\n&amp;\\text{\u91cf\u8bcd\u5185\u90e8\u7684\u4ee3\u6570\u8fd0\u7b97\u3002}&amp; \\\\\n\n(5)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k \\in \\mathbb{Z})(b+c= ka) &amp;\\text{; \u7531(4)}\\\\\n\n&amp;\\text{$\\mathbb{Z}$ \u5bf9\u52a0\u6cd5\u7684\u5c01\u95ed\u6027\u3002}&amp; \\\\\n\n(6)&amp;\\{a\\mid b, a\\mid c\\}\\vdash a\\mid (b+c) &amp;\\text{; \u53ef\u9664\u6027\u7684\u5b9a\u4e49 (5)}\\\\\n\n(7)&amp;\\vdash (a\\mid b \\wedge a\\mid c) \\rightarrow a\\mid (b+c) &amp;\\text{; $\\wedge$-\u6d88\u53bb (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{; \u7531(3)}\\\\\n\n&amp;\\text{\u91cf\u8bcd\u5185\u90e8\u7684\u4ee3\u6570\u8fd0\u7b97\u3002}&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{; \u7531(8)}\\\\\n\n&amp;\\text{$\\mathbb{Z}$ \u5bf9\u51cf\u6cd5\u7684\u5c01\u95ed\u6027\u3002}&amp; \\\\\n\n(10)&amp;\\{a\\mid b, a\\mid c\\}\\vdash a\\mid (b-c) &amp;\\text{; \u53ef\u9664\u6027\u7684\u5b9a\u4e49 (9)}\\\\\n\n(11)&amp;\\vdash (a\\mid b \\wedge a\\mid c) \\rightarrow a\\mid (b-c) &amp;\\text{; $\\wedge$-\u6d88\u53bb (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{; \u5728\u7ed3\u8bba\u4e2d\u8fdb\u884c $\\wedge$-\u5f15\u5165 (7,11) }\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}<\/span>\n<\/li>\n<li><strong>\u4e0e\u4e58\u79ef\u7684\u76f8\u5bb9\u6027\uff1a<\/strong> \u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>\uff0c\u5219\u5bf9\u4efb\u610f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in\\mathbb{Z}<\/span><\/span> \u90fd\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (bc)<\/span><\/span>\u3002<br \/>\n<u>\u8bc1\u660e<\/u>\uff1a<\/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{; \u53ef\u9664\u6027\u7684\u5b9a\u4e49\uff0c\u524d\u63d0}\\\\\n\n(2)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (\\exists k\\in\\mathbb{Z})(cb=cka) &amp;\\text{; \u7531(1)\uff0c$\\forall$-\u5f15\u5165\uff08$c$ \u4efb\u610f\uff09}\\\\\n\n&amp;\\text{\u91cf\u8bcd\u5185\u90e8\u7684 $\\mathbb{Z}$ \u4e2d\u7684\u4ee3\u6570\u8fd0\u7b97\u3002}&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{; \u7531(2)\uff0c\u5c01\u95ed\u6027\uff1a}\\overline{k}=ck\\\\\n\n(4)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (a \\mid cb) &amp;\\text{; \u53ef\u9664\u6027\u7684\u5b9a\u4e49 (3)}\\\\\n\n(5)&amp; \\vdash a\\mid b \\rightarrow \\left(\\forall c \\in \\mathbb{Z}\\right) (a \\mid cb) &amp;\\text{; \u63a8\u5bfc\u5b9a\u7406 (4)}\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<\/ul>\n<h3><b>\u5b9a\u7406\uff1a<\/b>\u9664\u6570\u7684\u754c<\/h3>\n<p style=\"text-align: justify;\">\n\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\neq 0<\/span><\/span> \u4e14 <span dir=\"ltr\"> <span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>\uff0c\u5219 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|\\le |b|<\/span><\/span>\u3002\n<\/p>\n<p><b>\u8bc1\u660e\uff1a<\/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{; \u524d\u63d0} \\\\\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{; \u53ef\u9664\u6027\u7684\u5b9a\u4e49\uff0c\u524d\u63d0} \\\\\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{; \u7edd\u5bf9\u503c\u7684\u6027\u8d28\uff0c\u7531(2)} \\\\\n\n(4) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash (\\exists k \\in \\mathbb{Z}) (k\\neq 0 \\wedge |b|=|k||a|)\n\n&amp; \\text{; \u7531(1,3)} \\\\\n\n(5) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash (\\exists k \\in \\mathbb{Z}) (1\\le |k| \\wedge |b|=|k||a|)\n\n&amp; \\text{; \u7531(4)\uff0c\u82e5 }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{; \u7531(5)} \\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<p><a name=\"3\"><\/a><\/br><\/p>\n<h2>\u4e60\u9898<\/h2>\n<ol>\n<li>\u8bc1\u660e\u5f53 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span> \u65f6\uff0c\u201c\u9664\u6570\u7684\u754c\u201d\u5b9a\u7406\u4e0d\u4e00\u5b9a\u6210\u7acb\u3002<\/li>\n<li>\u8003\u8651\u4e00\u4e2a\u96c6\u5408 <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u53ca\u5176\u4e0a\u7684\u4e00\u4e2a\u5173\u7cfb <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span>\u3002\u82e5\u5143\u7d20 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x,y\\in A<\/span><\/span> \u6ee1\u8db3 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u901a\u8fc7 <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> \u4e0e <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> \u76f8\u5173\uff0c\u5219\u8bb0\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\rho y<\/span><\/span><\/span>\u3002\u82e5\u5173\u7cfb <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> \u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\uff0c\u5219\u79f0\u5176\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u4e0a\u7684<strong>\u504f\u5e8f\u5173\u7cfb<\/strong>\uff1a\n<p>a)<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x\\in A) (x\\rho x)<\/span><\/span>\uff0c<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>\u3002<\/p>\n<p>\u8bc1\u660e\u53ef\u9664\u5173\u7cfb\u5728\u6574\u6570\u96c6\u5408\u4e0a\u6784\u6210\u4e00\u4e2a\u504f\u5e8f\u5173\u7cfb\u3002<\/li>\n<li>\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u8bc1\u660e\uff1a\u82e5 <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>\uff0c\u5219\u5bf9\u4efb\u610f\u96c6\u5408 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{x_i\\}_{i=1}^n \\subset \\mathbb{Z}<\/span><\/span> \u90fd\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid \\sum_{i=1}^n b_i x_i<\/span><\/span>\u3002\u6b64\u5916\uff0c\u8bc1\u660e\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b_i<\/span><\/span>\uff0c\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i\\in \\{1,2,3,\\cdots, n\\}<\/span><\/span>\uff0c\u4e14 <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> \u53ef\u4ee5\u8868\u793a\u4e3a\u8fd9\u4e9b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b_i<\/span><\/span> \u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u5219 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>\u3002<\/li>\n<li>\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>\uff0c\u8bc1\u660e\u96c6\u5408 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{x\\;:\\; d\\mid a\\}<\/span><\/span> \u662f\u4e00\u4e2a\u6709\u9650\u96c6\u3002<\/li>\n<li>\u56fa\u5b9a\u4e00\u4e2a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\in\\mathbb{Z}^+<\/span><\/span>\uff0c\u5e76\u4ee4\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>\u8bc1\u660e\uff1a<\/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\">\u82e5\u5c06 <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> \u7684\u5143\u7d20\u6309\u9012\u589e\u987a\u5e8f\u6392\u5217\uff1a<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>\uff0c\u5219\u76f8\u5e94\u7684\u5143\u7d20 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\mid d_i<\/span><\/span>\uff08\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i \\in \\{1,2,\\cdots, t\\}<\/span><\/span>\uff09\u6309\u9012\u51cf\u987a\u5e8f\u6392\u5217\u3002<\/li>\n<\/ol>\n<\/li>\n<li>\u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}^+<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ab=c<\/span><\/span>\u3002\u8bc1\u660e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\min\\{a,b\\}\\le \\sqrt{c}<\/span><\/span>\u3002<\/li>\n<li>\u82e5\u6574\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2\\mid n<\/span><\/span>\uff0c\u5219\u79f0\u5176\u4e3a\u5076\u6570\uff1b\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2\\nmid n<\/span><\/span>\uff0c\u5219\u79f0\u5176\u4e3a\u5947\u6570\u3002\u8bc1\u660e\u4ee5\u4e0b\u548c\u4e0e\u5dee\u7684\u6027\u8d28\uff1a\n<ol>\n<li type=\"a\">\u4e24\u4e2a\u5076\u6570\u7684\u548c\u662f\u5076\u6570\u3002<\/li>\n<li type=\"a\">\u4e24\u4e2a\u5947\u6570\u7684\u548c\u662f\u5076\u6570\u3002<\/li>\n<li type=\"a\">\u4e00\u4e2a\u5076\u6570\u4e0e\u4e00\u4e2a\u5947\u6570\u7684\u5dee\u662f\u5947\u6570\u3002<\/li>\n<\/ol>\n<\/li>\n<li>\u82e5 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u662f\u4e00\u4e2a\u4e0d\u7b49\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\pm 1<\/span><\/span> \u7684\u5947\u6570\uff0c\u8bc1\u660e <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u4e0d\u53ef\u80fd\u540c\u65f6\u6574\u9664\u4e24\u4e2a\u8fde\u7eed\u7684\u5076\u6570\u3002<\/li>\n<li>\u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b,n\\in\\mathbb{Z}<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a-b|\\lt |n|<\/span><\/span>\u3002\u8bc1\u660e <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u65e2\u4e0d\u80fd\u6574\u9664 <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span>\uff0c\u4e5f\u4e0d\u80fd\u6574\u9664 <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span>\u3002<\/li>\n<li>\u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{Z}<\/span><\/span>\u3002\u8bc1\u660e\uff1a\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>\u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{Z}<\/span><\/span> \u4e14\u5047\u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\neq 0<\/span><\/span>\u3002\u8bc1\u660e\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ac\\mid bc<\/span><\/span>\uff0c\u5219\u5fc5\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>\u3002<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u53ef\u9664\u6027 \u53ef\u9664\u6027\u662f\u6570\u8bba\u771f\u6b63\u7684\u8d77\u70b9\uff0c\u56e0\u4e3a\u5b83\u5c06\u6574\u6570\u8f6c\u5316\u4e3a\u4e00\u4e2a\u5177\u6709\u7ed3\u6784\u7684\u7cfb\u7edf\uff1a\u4f60\u4e0d\u518d\u5c06\u6570\u5b57\u4ec5\u4ec5\u89c6\u4e3a\u201c\u6570\u91cf\u201d\uff0c\u800c\u662f\u5c06\u5176\u770b\u4f5c\u5f7c\u6b64\u4e4b\u95f4\u53ef\u4ee5\u6216\u4e0d\u53ef\u4ee5\u76f8\u4e92\u5951\u5408\u7684\u7ec4\u6210\u90e8\u5206\u3002\u501f\u52a9\u5355\u4e00\u7684\u7ed3\u6784 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