{"id":35990,"date":"2026-01-10T18:40:43","date_gmt":"2026-01-10T18:40:43","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35990"},"modified":"2026-01-10T18:40:43","modified_gmt":"2026-01-10T18:40:43","slug":"%e6%ac%a7%e5%87%a0%e9%87%8c%e5%be%97%e9%99%a4%e6%b3%95%e7%ae%97%e6%b3%95","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/zh\/%e6%ac%a7%e5%87%a0%e9%87%8c%e5%be%97%e9%99%a4%e6%b3%95%e7%ae%97%e6%b3%95\/","title":{"rendered":"\u6b27\u51e0\u91cc\u5f97\u9664\u6cd5\u7b97\u6cd5"},"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>\u9664\u6cd5\u7b97\u6cd5<\/h1>\n<p style=\"text-align:center;\"><em>\u5728\u672c\u8bfe\u7a0b\u4e2d\uff0c\u6211\u4eec\u5c06\u628a<strong>\u9664\u6cd5\u7b97\u6cd5<\/strong>\u4f5c\u4e3a\u4e00\u4e2a\u539f\u7406\u52a0\u4ee5\u5c55\u5f00\uff0c\u8be5\u539f\u7406\u5f62\u5f0f\u5316\u5730\u523b\u753b\u4e86\u6574\u6570\u60c5\u5f62\u4e0b\u7684\u552f\u4e00\u5206\u89e3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span>\uff0c\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>\u3002\u9996\u5148\u8bc1\u660e\u5546\u4e0e\u4f59\u6570\u7684<strong>\u5b58\u5728\u6027<\/strong>\uff0c\u968f\u540e\u8bc1\u660e\u5176<strong>\u552f\u4e00\u6027<\/strong>\u3002\u6700\u540e\uff0c\u5c06\u89e3\u91ca\u4f59\u6570\u7684\u610f\u4e49\uff0c\u628a\u7406\u8bba\u4e0e\u4f5c\u4e3a\u8ba1\u7b97\u8fc7\u7a0b\u7684<strong>\u957f\u9664\u6cd5\u7b97\u6cd5<\/strong>\u8054\u7cfb\u8d77\u6765\uff0c\u5e76\u9884\u793a\u5176\u4e0e\u6a21\u7b97\u672f\u53ca\u8ba1\u7b97\u5e94\u7528\u4e4b\u95f4\u7684\u81ea\u7136\u8054\u7cfb\u3002<\/em><\/p>\n<p style=\"text-align:center;\"><strong>\u5b66\u4e60\u76ee\u6807<\/strong><\/p>\n<p>\u5b8c\u6210\u672c\u8bfe\u7a0b\u540e\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a<\/p>\n<ol>\n<li><strong>\u8bc6\u522b<\/strong>\u6574\u6570\u9664\u6cd5\u7684\u57fa\u672c\u6982\u5ff5\u4e0e\u89d2\u8272\uff08\u88ab\u9664\u6570\u3001\u9664\u6570\u3001\u5546\u3001\u4f59\u6570\uff09\uff0c\u4ee5\u53ca\u4f5c\u4e3a\u7cbe\u786e\u60c5\u5f62\u7684\u53ef\u9664\u6027\u6982\u5ff5\u3002<\/li>\n<li><strong>\u89e3\u91ca<\/strong>\u6b27\u51e0\u91cc\u5f97\u9664\u6cd5\u7b97\u6cd5\/\u5b9a\u7406\u7684\u8868\u8ff0\uff0c\u5305\u62ec\u7528\u4e8e\u786e\u5b9a\u4f59\u6570\u7684\u6761\u4ef6\uff08<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>\uff09\u53ca\u5176\u907f\u514d\u6b67\u4e49\u7684\u76ee\u7684\u3002<\/li>\n<li><strong>\u5e94\u7528<\/strong>\u6b27\u51e0\u91cc\u5f97\u5206\u89e3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span> \u5728\u5177\u4f53\u793a\u4f8b\u4e2d\u6c42\u51fa <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> \u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span>\uff0c\u5e76\u9a8c\u8bc1\u4f59\u6570\u7684\u754c\u9650\u3002<\/li>\n<li><strong>\u5206\u6790<\/strong>\u6839\u636e\u88ab\u9664\u6570\u4e0e\u9664\u6570\u7b26\u53f7\u5212\u5206\u7684\u60c5\u5f62\uff0c\u8bba\u8bc1\u4e3a\u4f55\u6b27\u51e0\u91cc\u5f97\u7ea6\u5b9a\u4fdd\u6301\u4f59\u6570\u975e\u8d1f\u3002<\/li>\n<li><strong>\u6267\u884c<\/strong>\u57fa\u4e8e\u4f4d\u503c\u8868\u793a\u7684\u957f\u9664\u6cd5\u7b97\u6cd5\uff0c\u4f5c\u4e3a\u4e00\u79cd\u673a\u68b0\u5316\u8fc7\u7a0b\u6765\u5f97\u5230 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> \u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span>\u3002<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><b><u>\u5185\u5bb9\u7d22\u5f15<\/u>:<\/b><br \/>\n<a href=\"#1\">\u9664\u6cd5\u4e0e\u53ef\u9664\u6027<\/a><br \/>\n<a href=\"#2\">\u6b27\u51e0\u91cc\u5f97\u9664\u6cd5\u5b9a\u7406<\/a><br \/>\n<a href=\"#3\">\u6b27\u51e0\u91cc\u5f97\u9664\u6cd5\u7684\u8bc1\u660e<\/a><br \/>\n<a href=\"#4\">\u5546\u4e0e\u4f59\u6570\u7684\u89e3\u91ca<\/a><br \/>\n<a href=\"#5\">\u957f\u9664\u6cd5\u7b97\u6cd5<\/a><br \/>\n<a href=\"#6\">\u7ed3\u8bba<\/a><br \/>\n<a href=\"#7\">\u5efa\u8bae\u4e0e\u5df2\u89e3\u4e60\u9898<\/a>\n<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u9664\u6cd5\u4e0e\u53ef\u9664\u6027<\/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>\u5728\u7b97\u672f\u4e2d\uff0c\u53ef\u9664\u6027\u63cf\u8ff0\u7684\u662f\u201c\u7cbe\u786e\u201d\u7684\u60c5\u5f62\uff1a<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> \u8868\u793a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u53ef\u4ee5\u6070\u597d\u5199\u6210 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u7684\u500d\u6570\u3002\u7136\u800c\uff0c\u5c3d\u7ba1\u4efb\u610f\u4e24\u4e2a\u6574\u6570\u4e4b\u95f4\u5e76\u4e0d\u603b\u662f\u5b58\u5728\u53ef\u9664\u6027\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u63d0\u51fa\u8fd9\u6837\u7684\u95ee\u9898\uff1a\u201c\u4e00\u4e2a\u6570\u5728\u53e6\u4e00\u4e2a\u6570\u4e2d\u80fd\u5bb9\u7eb3\u591a\u5c11\u6b21\uff1f\u5982\u679c\u4e0d\u80fd\u6070\u597d\u5bb9\u7eb3\uff0c\u5269\u4e0b\u7684\u662f\u4ec0\u4e48\uff1f\u201d\u6b63\u662f\u5728\u8fd9\u4e00\u80cc\u666f\u4e0b\u4ea7\u751f\u4e86<strong>\u9664\u6cd5\u7b97\u6cd5<\/strong>\uff0c\u5b83\u4fdd\u8bc1\u4efb\u610f\u6574\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u90fd\u53ef\u4ee5\u8868\u793a\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u7684\u4e00\u4e2a\u500d\u6570\u52a0\u4e0a\u4e00\u4e2a\u201c\u4f59\u6570\u201d\u3002<\/p>\n<p>\u4f8b\u5982\uff0c\u8003\u8651 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=3<\/span><\/span> \u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=16<\/span><\/span>\u3002\u663e\u7136\uff0c3 \u4e0d\u80fd\u6574\u9664 16\u3002\u7136\u800c\uff0c\u4ecd\u7136\u53ef\u4ee5\u4e3a\u8be5\u9664\u6cd5\u8fd0\u7b97\u627e\u5230\u4e00\u4e2a\u5546\u548c\u4e00\u4e2a\u4f59\u6570\uff0c\u56e0\u4e3a<\/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> \u4e00\u7ec4 16 \u4e2a\u7bb1\u5b50\u88ab\u5206\u6210\u6bcf\u7ec4 3 \u4e2a\uff0c\u7ed3\u679c\u5f97\u5230 5 \u7ec4\u7bb1\u5b50\uff0c\u5e76\u4e14\u5269\u4f59 1 \u4e2a<\/figcaption><\/figure>\n<p>\u56e0\u6b64\uff0c\u5c06 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=16<\/span><\/span> \u9664\u4ee5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=3<\/span><\/span>\uff08\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u4e3a<strong>\u88ab\u9664\u6570<\/strong>\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u4e3a<strong>\u9664\u6570<\/strong>\uff09\uff0c\u53ef\u8868\u793a\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">16\/3<\/span><\/span> \u6216 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">16\\div 3<\/span><\/span>\uff0c\u5176<strong>\u5546<\/strong>\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=5<\/span><\/span>\uff0c<strong>\u4f59\u6570<\/strong>\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=1<\/span><\/span>\u3002\u4e00\u822c\u800c\u8a00\uff0c<em>\u9664\u6cd5\u7b97\u6cd5<\/em>\u65ad\u8a00\uff1a\u5bf9\u4e8e\u4efb\u610f\u6574\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>\uff08\u9664\u6570\uff09\u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span>\uff08\u88ab\u9664\u6570\uff09\uff0c\u4e14\u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>\uff0c\u5b58\u5728\u552f\u4e00\u7684\u6574\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> \u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> \u4f7f\u5f97<\/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>\u5728\u8be5\u793a\u4f8b\u4e2d\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">16=3\\cdot 5+1<\/span><\/span>\uff0c\u4e14\u6761\u4ef6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span> \u5f97\u5230\u6ee1\u8db3\uff0c\u56e0\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le 1&lt;3<\/span><\/span>\u3002\u8fd9\u79cd\u8868\u8ff0\u5728\u9664\u6570\uff08\u6216\u88ab\u9664\u6570\uff09\u53ef\u80fd\u4e3a\u8d1f\u7684\u60c5\u51b5\u4e0b\u907f\u514d\u4e86\u6b67\u4e49\uff0c\u5e76\u4fdd\u8bc1\u4f59\u6570\u59cb\u7ec8\u975e\u8d1f\u4e14\u4e25\u683c\u5c0f\u4e8e\u9664\u6570\u7684\u7edd\u5bf9\u503c\u3002<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u6b27\u51e0\u91cc\u5f97\u9664\u6cd5\u5b9a\u7406<\/h2>\n<p>\u5e94\u7528\u9664\u6cd5\u7b97\u6cd5\u6240\u5f97\u7684\u7ed3\u679c\u79f0\u4e3a<strong>\u6b27\u51e0\u91cc\u5f97\u9664\u6cd5<\/strong>\uff0c\u5176\u4f9d\u636e\u5982\u4e0b\u6240\u793a\u7684\u7ed3\u679c\u3002<\/p>\n<p><b>\u5b9a\u7406\uff1a<\/b>\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\u5219\u5b58\u5728\u552f\u4e00\u7684 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> \u4f7f\u5f97<\/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>\u6574\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> \u79f0\u4e3a<strong>\u5546<\/strong>\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> \u79f0\u4e3a\u5c06 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u9664\u4ee5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u6240\u5f97\u7684<strong>\u4f59\u6570<\/strong>\u3002<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u6b27\u51e0\u91cc\u5f97\u9664\u6cd5\u7684\u8bc1\u660e<\/h2>\n<p>\u8be5\u8bc1\u660e\u5206\u4e3a\u4e24\u90e8\u5206\uff1a\u9996\u5148\u8bc1\u660e\u5546\u4e0e\u4f59\u6570\u7684\u5b58\u5728\u6027\uff1b\u968f\u540e\uff0c\u5728\u5176\u5b58\u5728\u7684\u524d\u63d0\u4e0b\uff0c\u8bc1\u660e\u5b83\u4eec\u7684\u552f\u4e00\u6027\u3002<\/p>\n<h3>\u5b58\u5728\u6027<\/h3>\n<p>\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>\uff0c\u5e76\u5b9a\u4e49 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d=|a|<\/span><\/span>\uff0c\u4ece\u800c\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d&gt;0<\/span><\/span>\u3002\u6211\u4eec\u9996\u5148\u8bc1\u660e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\ge 0<\/span><\/span> \u7684\u60c5\u5f62\uff1a\u5c06\u8bf4\u660e\u5b58\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> \u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq+r<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;d<\/span><\/span>\u3002\u968f\u540e\u518d\u5904\u7406 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b&lt;0<\/span><\/span> \u7684\u60c5\u5f62\u3002\u6700\u540e\uff0c\u7528 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span> \u53d6\u4ee3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d<\/span><\/span>\uff0c\u4ece\u800c\u5f97\u5230\u5f62\u5f0f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span> \u4e14\u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>\u3002<\/p>\n<p>\u5bf9\u4e8e\u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\in\\mathbb{Z}<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\ge 0<\/span><\/span> \u7684\u60c5\u5f62\uff0c\u6211\u4eec\u5efa\u7acb\u547d\u9898 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(b)<\/span><\/span>\uff1a\u201c\u5b58\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> \u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq+r<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;d<\/span><\/span>\u201d\u3002\u6211\u4eec\u5c06\u901a\u8fc7\u6570\u5b66\u5f52\u7eb3\u6cd5\u8bc1\u660e\uff0c\u5bf9\u4e8e\u6240\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\ge 0<\/span><\/span>\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(b)<\/span><\/span> \u90fd\u6210\u7acb\u3002<\/p>\n<p><strong>\u57fa\u4f8b\uff1a<\/strong>\u5f53 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span> \u65f6\uff0c\u53d6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=0<\/span><\/span> \u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=0<\/span><\/span>\uff0c\u5f97\u5230 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=d\\cdot 0+0<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le 0&lt;d<\/span><\/span>\u3002\u56e0\u6b64\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(0)<\/span><\/span> \u4e3a\u771f\u3002<\/p>\n<p><strong>\u5f52\u7eb3\u6b65\u9aa4\uff1a<\/strong>\u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\ge 0<\/span><\/span>\uff0c\u5e76\u5047\u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(k)<\/span><\/span> \u4e3a\u771f\u3002\u5219\u5b58\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> \u4f7f\u5f97<\/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>\u5bf9\u7b49\u5f0f\u4e24\u8fb9\u540c\u65f6\u52a0\u4e0a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span>\uff0c\u5f97\u5230<\/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>\u6b64\u5916\uff0c\u7531 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;d<\/span><\/span> \u53ef\u77e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1\\le d<\/span><\/span>\u3002\u56e0\u6b64\uff0c\u53ea\u53ef\u80fd\u51fa\u73b0 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1&lt;d<\/span><\/span> \u6216 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1=d<\/span><\/span> \u4e24\u79cd\u60c5\u51b5\u3002<\/p>\n<p><b>\u60c5\u5f62 1\uff1a<\/b>\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1&lt;d<\/span><\/span>\uff0c\u5b9a\u4e49 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q&#039;=q<\/span><\/span> \u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r&#039;=r+1<\/span><\/span>\u3002\u5219<\/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>\u60c5\u5f62 2\uff1a<\/b>\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1=d<\/span><\/span>\uff0c\u5219 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k+1=dq+d=d(q+1)+0<\/span><\/span>\u3002\u5b9a\u4e49 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q&#039;=q+1<\/span><\/span> \u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r&#039;=0<\/span><\/span>\uff0c\u5e76\u6ee1\u8db3<\/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>\u5728\u4e24\u79cd\u60c5\u5f62\u4e0b\uff0c\u6211\u4eec\u90fd\u6784\u9020\u4e86\u6574\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q&#039;,r&#039;<\/span><\/span>\uff0c\u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k+1=dq&#039;+r&#039;<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&#039;&lt;d<\/span><\/span>\u3002\u56e0\u6b64\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(k+1)<\/span><\/span> \u4e3a\u771f\u3002\u7531\u5f52\u7eb3\u6cd5\u53ef\u5f97\uff0c\u5bf9\u4e8e\u6240\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\ge 0<\/span><\/span>\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(b)<\/span><\/span> \u90fd\u6210\u7acb\u3002<\/p>\n<p>\u73b0\u5728\u8003\u8651 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b&lt;0<\/span><\/span> \u7684\u60c5\u5f62\u3002\u6b64\u65f6\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-b&gt;0<\/span><\/span>\u3002\u5c06\u4e0a\u8ff0\u7ed3\u679c\u5e94\u7528\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-b<\/span><\/span>\uff0c\u5b58\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> \u4f7f\u5f97<\/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>\u4e24\u8fb9\u540c\u4e58\u4ee5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-1<\/span><\/span>\uff0c\u5f97\u5230<\/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>\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=0<\/span><\/span>\uff0c\u53ea\u9700\u53d6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_1=-q<\/span><\/span> \u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r_1=0<\/span><\/span>\uff0c\u5219\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq_1+r_1<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r_1&lt;d<\/span><\/span>\u3002<\/p>\n<p>\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r&gt;0<\/span><\/span>\uff0c\u5b9a\u4e49 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_1=-q-1<\/span><\/span> \u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r_1=d-r<\/span><\/span>\u3002\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0&lt;r&lt;d<\/span><\/span>\uff0c\u53ef\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0&lt;d-r&lt;d<\/span><\/span>\uff0c\u5373 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r_1&lt;d<\/span><\/span>\u3002\u6b64\u5916\uff0c<\/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>\u56e0\u6b64\uff0c\u5bf9\u4e8e\u6240\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\in\\mathbb{Z}<\/span><\/span>\uff0c\u90fd\u5b58\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_1,r_1\\in\\mathbb{Z}<\/span><\/span>\uff0c\u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq_1+r_1<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r_1&lt;d<\/span><\/span>\u3002<\/p>\n<p>\u6700\u540e\uff0c\u56de\u987e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d=|a|<\/span><\/span>\u3002\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a&gt;0<\/span><\/span>\uff0c\u5219 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d=a<\/span><\/span>\uff0c\u7b49\u5f0f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq_1+r_1<\/span><\/span> \u53ef\u5199\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=aq_1+r_1<\/span><\/span>\uff0c\u5e76\u4e14\u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r_1&lt;|a|<\/span><\/span>\u3002<\/p>\n<p>\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a&lt;0<\/span><\/span>\uff0c\u5219 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d=-a<\/span><\/span>\uff0c\u5e76\u4e14 <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>\u3002\u5b9a\u4e49 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=-q_1<\/span><\/span> \u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=r_1<\/span><\/span>\uff0c\u5373\u53ef\u5f97\u5230<\/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>\u7531\u6b64\uff0c\u5bf9\u4e8e\u4efb\u610f\u6ee1\u8db3 <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> \u7684\u60c5\u5f62\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> \u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> \u7684\u5b58\u5728\u6027\u5f97\u8bc1\u3002<\/p>\n<h3>\u552f\u4e00\u6027<\/h3>\n<p>\u8bbe\u5b58\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,q&#039;,r,r&#039;\\in\\mathbb{Z}<\/span><\/span>\uff0c\u4f7f\u5f97<\/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>\u5bf9\u4e0a\u8ff0\u4e24\u4e2a\u7b49\u5f0f\u76f8\u51cf\uff0c\u5f97\u5230<\/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>\u7531\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r,r&#039;&lt;|a|<\/span><\/span>\uff0c\u53ef\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|r&#039;-r|&lt;|a|<\/span><\/span>\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5de6\u4fa7\u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u7684\u500d\u6570\uff0c\u56e0\u6b64\u4e5f\u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span> \u7684\u500d\u6570\u3002\u7edd\u5bf9\u503c\u4e25\u683c\u5c0f\u4e8e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span> \u7684 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span> \u7684\u552f\u4e00\u6574\u6570\u500d\u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span>\uff1b\u4e8b\u5b9e\u4e0a\uff0c\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a| m \\neq 0<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m\\in\\mathbb{Z}<\/span><\/span>\uff0c\u5219 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\big| |a| m \\big| \\ge |a| <\/span><\/span>\u3002\u56e0\u6b64\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r&#039;-r=0<\/span><\/span>\uff0c\u5373 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=r&#039;<\/span><\/span>\uff1b\u4ee3\u5165 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span>\uff0c\u53ef\u63a8\u51fa <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=q&#039;<\/span><\/span>\u3002\u7531\u6b64\u53ef\u77e5\uff0c\u8be5\u5206\u89e3\u662f\u552f\u4e00\u7684\u3002<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>\u5546\u4e0e\u4f59\u6570\u7684\u89e3\u91ca<\/h2>\n<ul>\n<li style=\"text-align: justify;\">\n    <strong>\u201c\u4f59\u6570\u201d\u7684\u542b\u4e49\u3002<\/strong> \u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span>\uff0c\u5219 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=b-qa<\/span><\/span> \u8868\u793a\u4ece <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u4e2d\u53bb\u9664\u500d\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">qa<\/span><\/span> \u540e\u6240\u201c\u5269\u4f59\u201d\u7684\u90e8\u5206\u3002\u6362\u8a00\u4e4b\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> \u662f\u5c06 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u8c03\u6574\u5230 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u7684\u500d\u6570\u7ec4\u6210\u7684\u7f51\u683c\u4e0a\u65f6\u6240\u4ea7\u751f\u7684\u5269\u4f59\u91cf\u3002\n  <\/li>\n<li style=\"text-align: justify;\">\n    <strong>\u4e3a\u4f55\u8981\u6c42<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>\u3002\u82e5\u6ca1\u6709\u8be5\u6761\u4ef6\uff0c\u6210\u5bf9\u7684 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(q,r)<\/span><\/span> \u5c06\u4e0d\u518d\u552f\u4e00\u3002\u4e8b\u5b9e\u4e0a\uff0c\u82e5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span>\uff0c\u5219\u540c\u6837\u53ef\u4ee5\u5199\u6210 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=a(q+1)+(r-a)<\/span><\/span>\u3002\u6761\u4ef6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span> \u5c06\u4f59\u6570\u9650\u5236\u5728\u56fa\u5b9a\u533a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{0,1,2,\\dots,|a|-1\\}<\/span><\/span> \u5185\uff0c\u4ece\u800c\u963b\u65ad\u4e86\u5bf9\u540c\u4e00\u6574\u6570\u8fdb\u884c\u65e0\u9650\u6b21\u201c\u91cd\u65b0\u6807\u8bb0\u201d\u7684\u53ef\u80fd\u6027\u3002\n  <\/li>\n<li style=\"text-align: justify;\">\n    <strong>\u5f53<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> <strong>\u4e3a\u8d1f\u65f6\u4f1a\u53d1\u751f\u4ec0\u4e48\u3002<\/strong> \u5b9a\u7406\u5728\u4e0d\u4f5c\u4efb\u4f55\u4fee\u6539\u7684\u60c5\u51b5\u4e0b\u4f9d\u7136\u6210\u7acb\uff1a\u4f59\u6570\u4fdd\u6301\u975e\u8d1f\uff0c\u5e76\u53d7 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span> \u7684\u4e0a\u754c\u7ea6\u675f\u3002\u8fd9\u4e00\u70b9\u5f88\u91cd\u8981\uff0c\u56e0\u4e3a\u67d0\u4e9b\u7f16\u7a0b\u8bed\u8a00\u91c7\u7528\u5411\u96f6\u622a\u65ad\uff0c\u53ef\u80fd\u4ea7\u751f\u8d1f\u4f59\u6570\uff1b\u800c\u5728\u6570\u5b66\u4e2d\u5219\u91c7\u7528\u7ea6\u5b9a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>\uff0c\u4f7f\u4f59\u6570\u6210\u4e3a\u4e00\u4e2a\u201c\u6807\u51c6\u201d\u7684\u4ee3\u8868\u3002<\/p>\n<p>    \u6b64\u5916\uff0c\u5f53 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a&gt;0<\/span><\/span> \u65f6\uff0c\u5546\u4e0e\u4e0b\u53d6\u6574\u51fd\u6570\u4e00\u81f4\uff1a<\/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>\u957f\u9664\u6cd5\u7b97\u6cd5<\/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>\u57fa\u4e8e\u6b27\u51e0\u91cc\u5f97\u9664\u6cd5\uff0c\u53ef\u4ee5\u5229\u7528\u6570\u5b57\u7684\u4f4d\u503c\u8868\u793a\u6765\u5b9e\u73b0\u957f\u9664\u6cd5\u7b97\u6cd5\u3002\u8be5\u7b97\u6cd5\u662f\u4e00\u79cd\u8ba1\u7b97\u6280\u672f\uff0c\u5f53\u9700\u8981\u8ba1\u7b97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\div a<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}<\/span><\/span> \u5e76\u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span> \u65f6\uff0c\u5b83\u80fd\u591f\u5feb\u901f\u6c42\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> \u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> \u7684\u503c\u3002<\/p>\n<p>\u4e3a\u63cf\u8ff0\u8be5\u7b97\u6cd5\uff0c\u6211\u4eec\u9996\u5148\u56de\u987e\u4e00\u4e9b\u793a\u4f8b\uff0c\u4ee5\u8bf4\u660e\u7b97\u6cd5\u6267\u884c\u8fc7\u7a0b\u4e2d\u7684\u6b65\u9aa4\u4ee5\u53ca\u53ef\u80fd\u51fa\u73b0\u7684\u60c5\u51b5\u3002<\/p>\n<ol>\n<li>\n<p>\u5047\u8bbe\u6211\u4eec\u8981\u8ba1\u7b97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">57\\div 4<\/span><\/span>\u3002\u5176\u4e2d 57 \u662f<strong>\u88ab\u9664\u6570<\/strong>\uff0c4 \u662f<strong>\u9664\u6570<\/strong>\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u5c06\u6267\u884c\u5982\u4e0b\u8ba1\u7b97\u5e8f\u5217\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; \\color{red}57 \\div 4 =\\color{black}  \\\\ &amp; \\text{\u5199\u51fa\u9664\u6cd5\u8fd0\u7b97\u3002} \\\\ \\\\\n\n(2) &amp; \\color{red}5&#039;\\color{black}7 \\div 4 =  \\\\ &amp; \\text{\u4ece\u5de6\u4fa7\u5206\u79bb\u51fa\u88ab\u9664\u6570\u7684\u7b2c\u4e00\u4e2a\u524d\u7f00\uff0c\u4f7f\u5176}\\\\\n\n&amp;\\text{\u5927\u4e8e\u6216\u7b49\u4e8e\u9664\u6570\uff0c\u672c\u4f8b\u4e2d\u4e3a 5\u3002} \\\\ \\\\\n\n(3) &amp; 5&#039;7 \\div 4 = \\color{red}1\\color{black} \\\\ &amp; \\text{\u601d\u8003\u4e00\u4e2a\u6700\u5927\u7684\u6570\uff0c\u4f7f\u5176\u4e0e $4$ \u76f8\u4e58\u540e}\\\\\n\n&amp;\\text{\u5c0f\u4e8e\u6216\u7b49\u4e8e $5$\uff0c\u5e76\u5c06\u5176\u5199\u5728\u7b49\u53f7\u53f3\u4fa7\u3002\u8be5\u6570\u4e3a 1\u3002} \\\\ \\\\\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{\u5c06\u7ed3\u679c\u4e58\u4ee5\u9664\u6570\uff0c\u5e76\u4ece\u6240\u9009\u6570\u4e2d\u51cf\u53bb\u3002} \\\\ \\\\\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{\u9009\u62e9\u5e76\u201c\u4e0b\u79fb\u201d\u4e0b\u4e00\u4e2a\u6570\u5b57\u3002} \\\\ \\\\\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{\u7528\u6700\u540e\u5f97\u5230\u7684\u6570\u91cd\u590d\u4e0a\u8ff0\u6b65\u9aa4\u3002}\n\n\\end{array}\n\n<\/span>\n<p>\u5f53\u6ca1\u6709\u66f4\u591a\u6570\u5b57\u53ef\u4ee5\u201c\u4e0b\u79fb\u201d\u65f6\uff0c\u7b97\u6cd5\u7ed3\u675f\uff0c\u6240\u5f97\u7ed3\u679c\u4e3a\u5546 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=14<\/span><\/span> \u548c\u4f59\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=1<\/span><\/span>\u3002\u7279\u522b\u5730\uff0c\u4f59\u6570\u59cb\u7ec8\u5c0f\u4e8e\u9664\u6570\u3002<\/p>\n<\/li>\n<li>\n<p>\u5047\u8bbe\u6211\u4eec\u8981\u8ba1\u7b97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">132\\div 5<\/span><\/span>\u3002\u4e3a\u6b64\uff0c\u5c06\u6267\u884c\u5982\u4e0b\u8ba1\u7b97\u5e8f\u5217\uff1a<\/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{\u5206\u79bb\u88ab\u9664\u6570\u7684\u7b2c\u4e00\u4e2a\u524d\u7f00\uff0c\u5e76\u5bfb\u627e\u4e00\u4e2a\u81ea\u7136\u6570\uff0c}\\\\\n\n&amp; \\text{\u4f7f\u5176\u4e0e 5 \u76f8\u4e58\u540e\u5c0f\u4e8e\u6216\u7b49\u4e8e\u8be5\u524d\u7f00\u3002\u82e5\u65e0\u6cd5\u505a\u5230\uff0c}\\\\\n\n&amp; \\text{\u5219\u5e76\u5165\u4e0b\u4e00\u4e2a\u6570\u5b57\uff0c\u76f4\u5230\u53ef\u4ee5\u4e3a\u6b62\u3002} \\\\ \\\\\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{\u4e00\u65e6\u4e0a\u8ff0\u6b65\u9aa4\u53ef\u884c\uff0c\u5373\u6309\u5e38\u89c4\u6267\u884c\u7b97\u6cd5\u3002} \\\\ \\\\\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>\u82e5\u88ab\u9664\u6570\u5177\u6709\u66f4\u591a\u4f4d\u6570\uff0c\u5219\u4ee5\u76f8\u540c\u65b9\u5f0f\u6267\u884c\u8be5\u8fc7\u7a0b\u3002\u4f8b\u5982\uff0c\u8ba1\u7b97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">3521\\div 12<\/span><\/span> \u65f6\uff0c\u5f97\u5230\uff1a<\/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>\u82e5\u5176\u4e2d\u4e00\u4e2a\u6570\u4e3a\u8d1f\uff0c\u5219\u5b58\u5728\u4e24\u79cd\u53ef\u63a5\u53d7\u7684\u7ed3\u679c\u8868\u793a\u65b9\u5f0f\u3002\u7136\u800c\uff0c\u5728\u6570\u8bba\u4e2d\u91c7\u7528\u7684\u6807\u51c6\u5f62\u5f0f\u662f\u4f59\u6570\u4e3a\u6b63\u7684\u8868\u793a\u65b9\u5f0f\uff0c\u4ece\u800c\u4e0e\u6b27\u51e0\u91cc\u5f97\u9664\u6cd5\u4fdd\u6301\u4e00\u81f4\u3002\u4f8b\u5982\uff0c\u82e5\u8ba1\u7b97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598\\div 21<\/span><\/span>\uff0c\u53ef\u5f97\u5230\u5982\u4e0b\u4e24\u79cd\u5c55\u5f00\u5f62\u5f0f\uff0c\u4e8c\u8005\u4f5c\u4e3a\u6052\u7b49\u5f0f\u5747\u662f\u6b63\u786e\u7684\uff1a<\/p>\n<ul>\n<li>\n        <b>\u5e26\u8d1f\u4f59\u6570\uff1a<\/b> \u8be5\u5f62\u5f0f\u901a\u8fc7\u5728\u8fd0\u7b97\u5f00\u59cb\u65f6\u53bb\u6389\u7b26\u53f7\uff0c\u5e76\u5728\u6700\u540e\u6062\u590d\u7b26\u53f7\u5f97\u5230\u3002<\/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>\u7531\u6b64\u53ef\u5f97\uff1a<\/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>\u968f\u540e\uff0c\u5c06\u6574\u4e2a\u7b49\u5f0f\u4e58\u4ee5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-1<\/span><\/span>\uff0c\u5f97\u5230\uff1a<\/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>\u56e0\u6b64\uff0c\u5728\u8be5\u8868\u793a\u4e0b\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598\\div 21<\/span><\/span> \u7684\u7ed3\u679c\u4e3a\u5546 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=-28<\/span><\/span>\uff0c\u4f59\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=-10<\/span><\/span>\u3002<\/p>\n<p>\u82e5\u518d\u6b21\u5c06\u6574\u4e2a\u7b49\u5f0f\u4e58\u4ee5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-1<\/span><\/span>\uff0c\u5219\u5f97\u5230\uff1a<\/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>\u56e0\u6b64\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">598\\div (-21)<\/span><\/span> \u7684\u7ed3\u679c\u4e3a\u5546 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=-28<\/span><\/span>\uff0c\u4f59\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=10<\/span><\/span>\u3002<\/p>\n<\/li>\n<li>\n        <b>\u5e26\u6b63\u4f59\u6570\u5e76\u4e0e\u6b27\u51e0\u91cc\u5f97\u9664\u6cd5\u4e00\u81f4\uff1a<\/b> \u57fa\u4e8e\u4e0a\u8ff0\u5c55\u5f00\uff0c\u53ef\u5f97\uff1a<\/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>\u968f\u540e\uff0c\u5728\u7b49\u5f0f\u53f3\u4fa7\u52a0\u4e0a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=21-21<\/span><\/span>\uff0c\u5f97\u5230\uff1a<\/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>\u56e0\u6b64\uff0c\u82e5\u4e0e\u6b27\u51e0\u91cc\u5f97\u9664\u6cd5\u4fdd\u6301\u4e00\u81f4\uff0c\u8ba1\u7b97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598\\div 21<\/span><\/span> \u7684\u7ed3\u679c\u4e3a\u5546 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=-29<\/span><\/span>\uff0c\u4f59\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=11<\/span><\/span>\u3002<\/p>\n<\/li>\n<\/ul>\n<p>\u5c31\u6b27\u51e0\u91cc\u5f97\u9664\u6cd5\u4e2d\u7684\u7ea6\u5b9a\u800c\u8a00\uff0c\u53ea\u6709\u4f59\u6570\u4e3a\u6b63\u7684\u7ed3\u679c\u624d\u5bf9\u5e94\u4e8e\u6b27\u51e0\u91cc\u5f97\u610f\u4e49\u4e0b\u7684\u5546\u4e0e\u4f59\u6570\u3002\u7136\u800c\uff0c\u8fd9\u4e24\u79cd\u8868\u8fbe\u5f62\u5f0f\u90fd\u662f\u6b63\u786e\u7684\u6052\u7b49\u5f0f\uff0c\u5e76\u4e14\u5728\u4e0d\u540c\u8bed\u5883\u4e2d\u53ef\u80fd\u5177\u6709\u5b9e\u7528\u4ef7\u503c\u3002\u5c3d\u7ba1\u957f\u9664\u6cd5\u7b97\u6cd5\u53ef\u80fd\u4ea7\u751f\u8d1f\u4f59\u6570\uff0c\u4f46\u8fd9\u5728\u673a\u68b0\u5316\u5730\u83b7\u5f97\u7ed3\u679c\u4ee5\u53ca\u8fdb\u884c\u4ee3\u6570\u8fd0\u7b97\u800c\u65e0\u9700\u989d\u5916\u6b65\u9aa4\u65f6\u662f\u4fbf\u5229\u7684\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u4e0e\u6b27\u51e0\u91cc\u5f97\u9664\u6cd5\u4e00\u81f4\u3001\u4f59\u6570\u4e3a\u6b63\u7684\u7ed3\u679c\u80fd\u591f\u4e3a\u6bcf\u4e00\u4e2a\u5269\u4f59\u7c7b\u786e\u5b9a\u4e00\u4e2a\u89c4\u8303\u4ee3\u8868\uff08\u4f8b\u5982\uff0c\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{0,1,\\dots,n-1\\}<\/span><\/span> \u4e2d\uff09\uff0c\u4ece\u800c\u4f7f\u6574\u6570\u7684\u6807\u8bb0\u66f4\u52a0\u7cbe\u786e\u4e14\u907f\u514d\u6b67\u4e49\u3002\u9700\u8981\u6307\u51fa\u7684\u662f\uff0c\u4e00\u65e6\u89c4\u5219\u88ab\u56fa\u5b9a\uff0c\u4e5f\u8fd8\u5b58\u5728\u5176\u4ed6\u540c\u6837\u6709\u6548\u7684\u4ee3\u8868\u7ea6\u5b9a\uff0c\u4f8b\u5982\u5bf9\u79f0\u4f59\u6570\u3002<\/p>\n<\/li>\n<\/ol>\n<p><a name=\"6\"><\/a><\/p>\n<h2>\u7ed3\u8bba<\/h2>\n<p>\u9664\u6cd5\u7b97\u6cd5\u4fdd\u8bc1\u4efb\u4f55\u6574\u6570\u9664\u6cd5\u90fd\u53ef\u4ee5\u552f\u4e00\u5730\u8868\u793a\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span>\uff0c\u5e76\u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>\uff0c\u4ece\u800c\u786e\u7acb\u4e86\u4e00\u4e2a\u6807\u51c6\u51c6\u5219\uff0c\u7528\u4ee5\u89e3\u91ca\u5373\u4f7f\u5728\u6d89\u53ca\u8d1f\u6570\u65f6\u7684\u5546\u4e0e\u4f59\u6570\u3002\u957f\u9664\u6cd5\u7b97\u6cd5\u4e0d\u8fc7\u662f\u8fd9\u79cd\u5206\u89e3\u5728\u5b9e\u8df5\u4e2d\u7684\u4e00\u79cd\u5b9e\u73b0\uff0c\u5b83\u4f9d\u6258\u4e8e\u4f4d\u503c\u8868\u793a\uff0c\u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> \u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> \u53ef\u4ee5\u901a\u8fc7\u673a\u68b0\u65b9\u5f0f\u8ba1\u7b97\u5f97\u5230\u3002\u6700\u540e\uff0c\u8fd9\u79cd\u6b27\u51e0\u91cc\u5f97\u5f62\u5f0f\u4e0d\u4ec5\u907f\u514d\u4e86\u6b67\u4e49\uff0c\u800c\u4e14\u8fd8\u81ea\u7136\u5730\u4e0e\u6a21\u7b97\u672f\u76f8\u8054\u7cfb\uff1a\u4f59\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> \u5145\u5f53\u4e86 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> \u6a21 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> \u7684\u5269\u4f59\u7c7b\u7684\u89c4\u8303\u4ee3\u8868\uff0c\u8fd9\u662f\u6570\u8bba\u4e2d\u8bb8\u591a\u6280\u672f\u4ee5\u53ca\u8ba1\u7b97\u5e94\u7528\u7684\u57fa\u7840\uff0c\u8fd9\u4e9b\u5185\u5bb9\u5c06\u5728\u540e\u7eed\u7ae0\u8282\u4e2d\u52a0\u4ee5\u8ba8\u8bba\u3002<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h2>\u4e60\u9898\uff08\u542b\u793a\u4f8b\u89e3\u7b54\uff09<\/h2>\n<h3>\u6b27\u51e0\u91cc\u5f97\u9664\u6cd5<\/h3>\n<ol>\n<li><strong>\uff08\u5df2\u89e3\uff09<\/strong> \u6c42 <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> \u4e0e <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span>\uff0c\u4f7f\u5f97\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>\u89e3\uff1a<\/strong>\u5bfb\u627e\u4e00\u4e2a\u6700\u5927\u7684\u6570\uff0c\u4f7f\u5176\u4e0e <span class=\"katex-eq\" data-katex-display=\"false\">9<\/span> \u76f8\u4e58\u540e\u4e0d\u8d85\u8fc7 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">137<\/span><\/span>\u3002\u6211\u4eec\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">15\\times 9 = 135<\/span><\/span>\uff0c\u800c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">16\\times 9 = 144<\/span><\/span>\uff0c\u56e0\u6b64\u6240\u6c42\u7684\u5546\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=15<\/span><\/span>\u3002\u63a5\u7740\u8ba1\u7b97\u4f59\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=137-9q<\/span><\/span>\uff0c\u5e76\u9a8c\u8bc1 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r\\lt 9<\/span><\/span>\u3002\u8ba1\u7b97\u5f97\u5230\uff1a<\/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>\u56e0\u6b64\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">137=9\\cdot 15+2<\/span><\/span>\uff0c\u5e76\u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le 2\\lt 9<\/span><\/span>\u3002<\/p>\n<\/li>\n<li><strong>\uff08\u5df2\u89e3\uff09<\/strong> \u6c42 <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> \u4e0e <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span>\uff0c\u4f7f\u5f97\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>\u89e3\uff1a<\/strong>\u5bfb\u627e\u4e00\u4e2a\u6700\u5927\u7684\u6570\uff0c\u4f7f\u5176\u4e0e <span class=\"katex-eq\" data-katex-display=\"false\">37<\/span> \u76f8\u4e58\u540e\u4e0d\u8d85\u8fc7 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2025<\/span><\/span>\u3002\u6ce8\u610f\u5230 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">37\\times 54 = 1998<\/span><\/span>\uff0c\u800c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">37\\times 55 = 2035<\/span><\/span>\uff0c\u56e0\u6b64\u6240\u6c42\u7684\u5546\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=54<\/span><\/span>\u3002\u63a5\u7740\u8ba1\u7b97\u4f59\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=2025-37q<\/span><\/span>\uff0c\u5e76\u9a8c\u8bc1 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r\\lt 37<\/span><\/span>\u3002\u8ba1\u7b97\u5f97\u5230\uff1a<\/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>\u56e0\u6b64\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2025=37\\cdot 54+27<\/span><\/span>\uff0c\u5e76\u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le 27\\lt 37<\/span><\/span>\u3002<\/p>\n<\/li>\n<li>\u6c42 <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> \u4e0e <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span>\uff0c\u4f7f\u5f97\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>\u6c42 <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> \u4e0e <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span>\uff0c\u4f7f\u5f97\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>\u6c42 <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> \u4e0e <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span>\uff0c\u4f7f\u5f97\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>\u56fa\u5b9a\u4f59\u6570\u4e0e\u6574\u6570\u7684\u5bfb\u627e\u3002<\/strong> \u8bbe <span class=\"katex-eq\" data-katex-display=\"false\">a=12<\/span>\u3002\n<p>(a) \u63cf\u8ff0\u6240\u6709\u6574\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> \u7684\u96c6\u5408\uff0c\u4f7f\u5176\u9664\u4ee5 <span class=\"katex-eq\" data-katex-display=\"false\">12<\/span> \u7684\u6b27\u51e0\u91cc\u5f97\u9664\u6cd5\u4f59\u6570\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">r=5<\/span>\u3002<\/p>\n<p>(b) \u6c42\u6ee1\u8db3\u4e0a\u8ff0\u6761\u4ef6\u4e14 <span class=\"katex-eq\" data-katex-display=\"false\">b\\gt 1000<\/span> \u7684\u6700\u5c0f\u6574\u6570\uff0c\u5e76\u786e\u5b9a\u76f8\u5e94\u7684\u5546 <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span>\u3002<\/p>\n<\/li>\n<\/ol>\n<h3>\u957f\u9664\u6cd5\u7b97\u6cd5<\/h3>\n<p>\u5728\u6bcf\u4e00\u79cd\u60c5\u5f62\u4e2d\uff0c\u5e94\u7528\u957f\u9664\u6cd5\u7b97\u6cd5\u6765\u8ba1\u7b97\u5546 <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> \u548c\u4f59\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span>\u3002\u5f53\u7b97\u6cd5\u7ed9\u51fa\u8d1f\u4f59\u6570\u65f6\uff0c\u5c06\u7ed3\u679c\u8f6c\u6362\u4e3a\u6b27\u51e0\u91cc\u5f97\u9664\u6cd5\u7684\u5f62\u5f0f\u3002\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>\u9664\u6cd5\u7b97\u6cd5 \u5728\u672c\u8bfe\u7a0b\u4e2d\uff0c\u6211\u4eec\u5c06\u628a\u9664\u6cd5\u7b97\u6cd5\u4f5c\u4e3a\u4e00\u4e2a\u539f\u7406\u52a0\u4ee5\u5c55\u5f00\uff0c\u8be5\u539f\u7406\u5f62\u5f0f\u5316\u5730\u523b\u753b\u4e86\u6574\u6570\u60c5\u5f62\u4e0b\u7684\u552f\u4e00\u5206\u89e3 \uff0c\u5176\u4e2d 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