{"id":113998,"date":"2026-10-08T07:50:03","date_gmt":"2026-10-07T23:50:03","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/113998.html"},"modified":"2026-10-08T07:50:03","modified_gmt":"2026-10-07T23:50:03","slug":"%e3%80%90%e4%be%8b-5%e3%80%91strange-way-to-express-integers%ef%bc%88%e4%bf%a1%e6%81%af%e5%ad%a6%e5%a5%a5%e8%b5%9b%e4%b8%80%e6%9c%ac%e9%80%9a-p1635%ef%bc%89","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/113998.html","title":{"rendered":"\u3010\u4f8b 5\u3011Strange Way to Express Integers\uff08\u4fe1\u606f\u5b66\u5965\u8d5b\u4e00\u672c\u901a- P1635\uff09"},"content":{"rendered":"

\u3010\u9898\u76ee\u63cf\u8ff0\u3011<\/span><\/h4>\n

\u539f\u9898\u6765\u81ea：POJ 2891<\/span><\/span><\/p>\n

\u7ed9\u5b9a\u00a02n\u00a0\u4e2a\u6b63\u6574\u6570\u00a0a1,a2,\u22ef,an\u548c\u00a0m1,m2,\u22ef,m[n] ，\u6c42\u4e00\u4e2a\u6700\u5c0f\u7684\u6b63\u6574\u6570\u00a0x，\u6ee1\u8db3\u00a0\u2200i\u2208[1,n],x\u2261ai(mod m[i])，\u6216\u8005\u7ed9\u51fa\u65e0\u89e3\u3002<\/span><\/span><\/p>\n

\u3010\u8f93\u5165\u3011<\/span><\/h4>\n

\u591a\u7ec4\u6570\u636e\u3002<\/span><\/span><\/p>\n

\u6bcf\u7ec4\u6570\u636e\u7b2c\u4e00\u884c\u4e00\u4e2a\u6574\u6570\u00a0n；<\/span><\/span><\/p>\n

\u63a5\u4e0b\u6765\u00a0n\u00a0\u884c，\u6bcf\u884c\u4e24\u4e2a\u6574\u6570\u00a0mi,ai\u00a0\u3002<\/span><\/span><\/p>\n

\u3010\u8f93\u51fa\u3011<\/span><\/h4>\n

\u5bf9\u4e8e\u6bcf\u7ec4\u6570\u636e，\u82e5\u65e0\u89e3，\u8f93\u51fa\u00a0\u22121；\u5426\u5219\u8f93\u51fa\u4e00\u4e2a\u975e\u8d1f\u6574\u6570，\u82e5\u6709\u591a\u89e3，\u8f93\u51fa\u6700\u5c0f\u7684\u6ee1\u8db3\u6761\u4ef6\u7684\u7b54\u6848\u3002<\/span><\/span><\/p>\n

\u3010\u8f93\u5165\u6837\u4f8b\u3011<\/span><\/h4>\n

2
\n8 7
\n11 9<\/span><\/p>\n

\u3010\u8f93\u51fa\u6837\u4f8b\u3011<\/span><\/h4>\n

31<\/span><\/p>\n

\u3010\u63d0\u793a\u3011<\/span><\/h4>\n

\u6570\u636e\u8303\u56f4\u4e0e\u63d0\u793a：<\/span><\/span><\/p>\n

\u5bf9\u4e8e\u5168\u90e8\u6570\u636e，\u6240\u6709\u7684\u8f93\u5165\u90fd\u662f\u975e\u8d1f\u7684，\u5e76\u4e14\u53ef\u4ee5\u7528\u00a064\u00a0\u4f4d\u6709\u7b26\u53f7\u6574\u6570\u8868\u793a\u3002\u4fdd\u8bc1\u00a01\u2264n\u226410^5,m[i]>a[i] \u3002<\/span><\/span><\/p>\n<\/p>\n

1. \u9898\u610f\u8f6c\u6362<\/h4>\n

\u7528\u4e00\u53e5\u8bdd\u5265\u79bb\u9898\u76ee\u7684\u80cc\u666f\u6545\u4e8b，\u8fd9\u9053\u9898\u672c\u8d28\u4e0a\u662f\u5728\u6c42：\u7ed9\u5b9a n \u4e2a\u7ebf\u6027\u540c\u4f59\u65b9\u7a0b x \u2261 a[i]\u200b (mod m[i]\u200b)，\u5728\u4e0d\u4fdd\u8bc1\u6a21\u6570m[i]\u4e24\u4e24\u4e92\u8d28\u7684\u60c5\u51b5\u4e0b，\u6c42\u51fa\u6700\u5c0f\u7684\u6b63\u6574\u6570\u89e3 x\u3002 \u8fd9\u6b63\u662f\u6570\u8bba\u4e2d\u5927\u540d\u9f0e\u9f0e\u7684\u7ec8\u6781\u6b66\u5668\u2014\u2014\u6269\u5c55\u4e2d\u56fd\u5269\u4f59\u5b9a\u7406（EXCRT） \u7684\u6807\u51c6\u6a21\u677f\u9898\u3002\u8fd9\u9053\u9898\u662f\u6570\u8bba\u8fdb\u9636\u9636\u6bb5\u7684\u7edd\u4f73\u8bd5\u91d1\u77f3\u3002<\/p>\n

2. \u601d\u8003\u8fc7\u7a0b\u4e0e\u89e3\u9898\u601d\u8def<\/h4>\n

\u9762\u5bf9\u65b9\u7a0b\u7ec4，\u6211\u4eec\u7684\u5927\u8111\u901a\u5e38\u4f1a\u7ecf\u5386\u4ee5\u4e0b\u8fed\u4ee3\u8fc7\u7a0b：<\/p>\n

    \n
  • \n

    \u7b2c\u4e00\u76f4\u89c9：\u66b4\u529b\u679a\u4e3e \u6700\u6734\u7d20\u7684\u60f3\u6cd5\u662f，\u4ece x=a[1]\u200b \u5f00\u59cb，\u6bcf\u6b21\u52a0\u4e0a m[1]\u200b，\u7136\u540e\u53bb\u8bd5\u662f\u5426\u6ee1\u8db3\u7b2c\u4e8c\u4e2a\u65b9\u7a0b x\u2261a[2]\u200b(mod m[2]\u200b)，\u6ee1\u8db3\u540e\u518d\u53bb\u8bd5\u7b2c\u4e09\u4e2a\u2026\u2026 \u4e3a\u4ec0\u4e48\u4f1a\u8d85\u65f6？ \u9898\u76ee\u4e2d n\u226410^5，\u6a21\u6570\u867d\u7136\u9898\u76ee\u8bf4 64 \u4f4d\u6574\u578b\u88c5\u5f97\u4e0b，\u4f46\u5168\u90e8\u4e58\u8d77\u6765（\u5373\u603b\u5468\u671f）\u662f\u4e00\u4e2a\u5929\u6587\u6570\u5b57\u3002\u9760\u52a0\u6cd5\u53bb\u904d\u5386\u72b6\u6001\u7a7a\u95f4，\u8bc4\u6d4b\u673a\u8dd1\u5230\u5b87\u5b99\u6bc1\u706d\u4e5f\u7b97\u4e0d\u5b8c\u3002<\/p>\n<\/li>\n

  • \n

    \u964d\u7ef4\u6253\u51fb：\u4e24\u4e24\u5408\u5e76，\u5316\u7e41\u4e3a\u7b80 \u65e2\u7136\u4e0d\u80fd\u5b8f\u89c2\u901a\u6740，\u6211\u4eec\u5c31\u201c\u5fae\u89c2\u8695\u98df\u201d\u3002\u5047\u8bbe\u6211\u4eec\u5df2\u7ecf\u5904\u7406\u5b8c\u4e86\u524d i\u22121 \u4e2a\u65b9\u7a0b，\u6c42\u51fa\u4e86\u4e00\u4e2a\u80fd\u540c\u65f6\u6ee1\u8db3\u5b83\u4eec\u7684\u7279\u89e3 ans\u3002\u540c\u65f6，\u524d i\u22121 \u4e2a\u65b9\u7a0b\u4f1a\u5f62\u6210\u4e00\u4e2a\u516c\u5171\u5927\u5468\u671f lcm\u3002 \u90a3\u4e48，\u524d i\u22121 \u4e2a\u65b9\u7a0b\u7684\u201c\u901a\u89e3\u201d\u5c31\u53ef\u4ee5\u5199\u6210：x=ans+t\u22c5lcm （t \u662f\u672a\u77e5\u7684\u6574\u6570\u6b65\u6570）\u3002<\/p>\n

    \u73b0\u5728，\u6211\u4eec\u8981\u8ba9\u8fd9\u4e2a\u901a\u89e3\u4e5f\u6ee1\u8db3\u7b2c i \u4e2a\u65b9\u7a0b：<\/p>\n

    ans+t\u22c5lcm\u2261a[i]\u200b (mod m[i]\u200b)<\/p>\n

    \u7a0d\u5fae\u79fb\u4e00\u4e0b\u9879，\u628a\u672a\u77e5\u6570\u653e\u5de6\u8fb9，\u5df2\u77e5\u5e38\u6570\u653e\u53f3\u8fb9：<\/p>\n

    t\u22c5lcm\u2212y\u22c5m[i]\u200b=a[i]\u200b\u2212ans<\/p>\n

    \u4ed4\u7ec6\u89c2\u5bdf，\u8fd9\u4e0d\u5c31\u662f\u4e00\u4e2a\u6807\u51c6\u7684\u4e8c\u5143\u4e00\u6b21\u4e0d\u5b9a\u65b9\u7a0b Ax+By=C \u5417？\u5176\u4e2d A=lcm，B=m[i\u200b]，C=a[i]\u200b\u2212ans\u3002\u672a\u77e5\u6570\u662f\u6211\u4eec\u60f3\u6c42\u7684\u6b65\u6570 t（\u5bf9\u5e94\u4ee3\u7801\u91cc\u7684 x0\u200b）\u3002 \u8ff7\u96fe\u5f7b\u5e95\u6563\u5f00\u2014\u2014\u76f4\u63a5\u638f\u51fa\u6269\u5c55\u6b27\u51e0\u91cc\u5f97\u7b97\u6cd5\u6c42\u89e3\u5373\u53ef\u3002<\/p>\n<\/li>\n<\/ul>\n

    3. \u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u6837\u4f8b\u63a8\u6f14<\/h4>\n

    \u6807\u7a0b\u7684\u6838\u5fc3\u7b97\u6cd5\u662f exgcd\u00a0+ \u8fed\u4ee3\u5408\u5e76\u3002 \u6211\u4eec\u7ef4\u62a4\u4e24\u4e2a\u6838\u5fc3\u53d8\u91cf：<\/p>\n

  • \n

    ans：\u5f53\u524d\u5df2\u7ecf\u5408\u5e76\u6210\u529f\u7684\u6240\u6709\u65b9\u7a0b\u7684\u6700\u5c0f\u6b63\u6574\u6570\u89e3（\u57fa\u5e95）\u3002<\/p>\n<\/li>\n

  • \n

    lcm：\u5f53\u524d\u5df2\u7ecf\u5408\u5e76\u6210\u529f\u7684\u6240\u6709\u65b9\u7a0b\u7684\u6700\u5c0f\u516c\u500d\u6570（\u5927\u5468\u671f）\u3002<\/p>\n<\/li>\n

    \u6838\u5fc3\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u5f0f：<\/p>\n

  • \n

    \u7b97\u5e38\u6570：C=( (a[i]\u200b\u2212ans) (mod m[i]\u200b) + m[i]\u200b) (mod m[i]\u200b)<\/p>\n<\/li>\n

  • \n

    \u6269\u6b27\u89e3\u57fa\u5e95：lcm\u22c5x0\u200b+m[i]\u200b\u22c5y0\u200b=d=gcd(lcm,m[i]\u200b)<\/p>\n<\/li>\n

  • \n

    \u5224\u65e0\u89e3：\u6839\u636e\u88f4\u8700\u5b9a\u7406，\u5982\u679c C \u4e0d\u80fd\u88ab d \u6574\u9664，\u76f4\u63a5 return -1\u3002<\/p>\n<\/li>\n

  • \n

    \u6269\u500d\u6570\u6c42\u771f\u5b9e\u6b65\u957f：t=(x0\u200b\u22c5C\/d)(mod m[i]\/d)<\/p>\n<\/li>\n

  • \n

    \u66f4\u65b0\u5168\u5c40\u89e3：ans=(ans+t\u22c5lcm) (mod next_lcm)<\/p>\n<\/li>\n

    \u6781\u7b80\u6570\u636e\u624b\u73a9\u63a8\u6f14（\u5e26\u5165\u6837\u4f8b\u6570\u636e）： \u65b9\u7a0b 1：x\u22617 (mod 8) \u65b9\u7a0b 2：x\u22619 (mod 11)<\/p>\n

      \n
    • \n

      \u521d\u59cb\u72b6\u6001：ans = 7, lcm = 8<\/p>\n<\/li>\n

    • \n

      \u63a5\u5165\u65b9\u7a0b 2：<\/p>\n

        \n
      • \n

        \u5e38\u6570 C=9\u22127=2\u3002<\/p>\n<\/li>\n

      • \n

        \u8c03\u7528 exgcd(8, 11, x0, y0)：\u7b97\u51fa 8\u22c5(\u22124)+11\u22c53=1\u3002\u5f97\u51fa d=1, x0=\u22124\u3002<\/p>\n<\/li>\n

      • \n

        \u8ba1\u7b97\u5c40\u90e8\u5468\u671f：mod=m[i]\u200b\/d=11\/1=11\u3002<\/p>\n<\/li>\n

      • \n

        \u6b65\u6570\u6269\u5927\u5e76\u8f6c\u6b63：x0=(\u22124\u00d72)(mod 11)\u2261\u22128\u22613 (mod 11)\u3002\u6240\u4ee5\u6211\u4eec\u8981\u8de8 3 \u6b65\u3002<\/p>\n<\/li>\n

      • \n

        \u8ba1\u7b97\u4e0b\u4e00\u8f6e\u603b\u5468\u671f：next_lcm=8\/1\u00d711=88\u3002<\/p>\n<\/li>\n

      • \n

        \u66f4\u65b0\u5168\u5c40\u89e3：ans=(7+3\u00d78) (mod 88)=31\u3002<\/p>\n<\/li>\n<\/ul>\n<\/li>\n

      • \n

        \u7ed3\u679c：\u8f93\u51fa 31，\u4e0e\u6837\u4f8b\u5b8c\u7f8e\u5951\u5408\u3002<\/p>\n<\/li>\n<\/ul>\n

        4. \u65f6\u7a7a\u590d\u6742\u5ea6\u5206\u6790<\/h4>\n
          \n
        • \n

          \u65f6\u95f4\u590d\u6742\u5ea6：O(nlog(max M))\u3002\u7b97\u6cd5\u5171\u6709 n \u8f6e\u5faa\u73af，\u6bcf\u6b21\u5faa\u73af\u6267\u884c\u4e00\u6b21 exgcd\u3002\u6269\u5c55\u6b27\u51e0\u91cc\u5f97\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u5bf9\u6570\u7ea7\u522b\u7684，\u6574\u4f53\u8dd1\u6ee1 10^5 \u6b21\u5927\u7ea6\u4ec5\u9700\u5341\u51e0\u6beb\u79d2，\u5bf9\u4e8e 1s \u7684\u65f6\u9650\u7ef0\u7ef0\u6709\u4f59\u3002<\/p>\n<\/li>\n

        • \n

          \u7a7a\u95f4\u590d\u6742\u5ea6：O(n)\u3002\u4ec5\u9700\u8981\u5f00\u4e24\u4e2a\u957f\u5ea6\u4e3a 100010 \u7684 long long \u6570\u7ec4\u5b58\u50a8\u8f93\u5165\u7684 a \u548c m，\u5185\u5b58\u6d88\u8017\u4e0d\u8db3 2\u00a0MB\u3002<\/p>\n<\/li>\n<\/ul>\n

          5. \u6613\u9519\u603b\u7ed3<\/h4>\n

          \u5f88\u591a\u540c\u5b66\u5728\u624b\u63a8\u6269\u5c55\u4e2d\u56fd\u5269\u4f59\u5b9a\u7406\u65f6，\u7ecf\u5e38\u88ab\u4ee3\u7801\u91cc\u6ee1\u5929\u98de\u7684\u53d6\u6a21\u641e\u6655\u3002\u4ec0\u4e48\u65f6\u5019\u8be5 % mod，\u4ec0\u4e48\u65f6\u5019\u8be5 % lcm？ \u8fd9\u662f\u4e00\u4e2a\u6781\u5176\u654f\u9510\u4e14\u76f4\u51fb\u672c\u8d28\u7684\u95ee\u9898！\u8fd9\u4e24\u7c7b\u53d6\u6a21\u7684\u4f5c\u7528\u5bf9\u8c61\u5b8c\u5168\u4e0d\u540c：<\/p>\n

          \u7075\u9b42\u62f7\u95ee 1：\u4e3a\u4ec0\u4e48\u6c42 x0\u200b \u65f6\u8981 % mod？<\/h5>\n

          % mod \u662f\u7528\u6765\u7ea6\u675f\u500d\u6570（\u7cfb\u6570 x0）\u7684\u3002 \u5728\u7b2c i \u8f6e\u65f6，\u4e3a\u4e86\u8ba9\u65b0\u65b9\u7a0b\u6210\u7acb，\u6211\u4eec\u79fb\u9879\u5f97\u5230\u4e86\u65b9\u7a0b：<\/p>\n

          x0\u200b\u22c5lcm+y0\u22c5m[i]\u200b=c<\/p>\n

          \u6b64\u65f6，\u6211\u4eec\u6c42\u51fa\u4e86\u6700\u5927\u516c\u7ea6\u6570 d=gcd(lcm,m[i]\u200b)\u3002\u628a\u65b9\u7a0b\u4e24\u8fb9\u540c\u65f6\u9664\u4ee5 d，\u8f6c\u6362\u56de\u540c\u4f59\u65b9\u7a0b：<\/p>\n

          x0\u22c5lcm\/d\u2261c\/d\u200b (mod m[i]\/d\u200b\u200b)<\/p>\n

          \u5728\u8fd9\u4e2a\u5316\u7b80\u540e\u7684\u65b9\u7a0b\u91cc，x0 \u7684\u5468\u671f（\u4e5f\u5c31\u662f\u65b0\u7684\u6a21\u6570）\u53d8\u6210\u4e86m[i]\/d\u3002\u8fd9\u6b63\u662f\u4ee3\u7801\u4e2d ll mod = m[i] \/ d; \u7684\u7531\u6765\u3002\u6211\u4eec\u5bf9 x0\u200b \u8fdb\u884c % mod \u64cd\u4f5c，\u662f\u4e3a\u4e86\u5728\u5c40\u90e8\u65b9\u7a0b\u91cc\u627e\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u6700\u5c0f\u6b63\u500d\u6570，\u9632\u6b62 x0 \u53d8\u5f97\u65e0\u7a77\u5927\u5bfc\u81f4\u540e\u7eed\u4e58\u6cd5\u6ea2\u51fa\u3002<\/p>\n

          \u7075\u9b42\u62f7\u95ee 2：\u4e3a\u4ec0\u4e48\u66f4\u65b0\u7b54\u6848\u65f6\u8981 % lcm (\u6216 % nextlcm)？<\/h5>\n

          % lcm \u662f\u7528\u6765\u7ea6\u675f\u771f\u5b9e\u7b54\u6848（ans）\u7684\u3002 \u5f53\u6211\u4eec\u6c42\u51fa\u4e86\u6700\u5c0f\u6b63\u500d\u6570 x0\u00a0\u540e，\u5c31\u7b97\u51fa\u4e86\u6ee1\u8db3\u524d i \u4e2a\u65b9\u7a0b\u7684\u65b0\u89e3：ans[new\u200b]=ans+x0\u200b\u22c5lcm\u3002 \u4f46\u8fd9\u53ea\u662f\u65e0\u7a77\u591a\u4e2a\u5408\u6cd5\u89e3\u4e2d\u7684\u4e00\u4e2a\u3002\u524d i \u4e2a\u65b9\u7a0b\u5408\u5e76\u540e，\u5b83\u4eec\u4f1a\u6709\u4e00\u4e2a\u5168\u5c40\u7684\u603b\u5faa\u73af\u5468\u671f\u3002\u8fd9\u4e2a\u5468\u671f\u5c31\u662f\u539f\u672c\u7684 lcm \u548c\u5f53\u524d\u6a21\u6570 mi\u200b \u7684\u6700\u5c0f\u516c\u500d\u6570，\u5373：<\/p>\n

          nextlcm=lcm\u22c5m[i]\/d\u200b\u200b<\/p>\n

          \u56e0\u4e3a\u9898\u76ee\u8981\u6c42\u7684\u662f\u6700\u5c0f\u6b63\u6574\u6570\u89e3，\u6240\u4ee5\u65e0\u8bba ans[new]\u200b\u53d8\u5230\u591a\u5927，\u5b83\u76f8\u5bf9\u4e8e\u65b0\u7684\u5927\u5468\u671f nextlcm \u7684\u4f59\u6570\u624d\u662f\u6700\u5c0f\u7684\u5408\u6cd5\u7b54\u6848\u3002<\/p>\n

          \u6838\u5fc3\u603b\u7ed3\u5bf9\u7167\u8868：<\/p>\n\n\u53d8\u91cf\u7269\u7406\u610f\u4e49\u6240\u5728\u7684\u65b9\u7a0b\u7a7a\u95f4\u5bf9\u5e94\u7684\u6a21\u6570\u4ee3\u7801\u4e2d\u7684\u53d8\u91cf\u540d<\/tr>\n\n\n\n
          x0\u200b<\/td>\n\u9700\u8981\u8de8\u8d8a\u591a\u5c11\u6b65\u624d\u80fd\u6ee1\u8db3\u5f53\u524d\u65b9\u7a0b<\/td>\n\u5c40\u90e8\u7684\u500d\u6570\u7a7a\u95f4<\/td>\nm[i]\u200b\/d<\/td>\nmod<\/td>\n<\/tr>\n
          ans<\/td>\n\u6ee1\u8db3\u5f53\u524d\u6240\u6709\u65b9\u7a0b\u7684\u5177\u4f53\u6570\u503c\u89e3<\/td>\n\u5168\u5c40\u7684\u771f\u5b9e\u503c\u7a7a\u95f4<\/td>\nlcm \u4e0e m[i]\u200b \u7684\u6700\u5c0f\u516c\u500d\u6570<\/td>\nnextlcm \u6216\u66f4\u65b0\u540e\u7684 lcm<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

          \u4e00\u53e5\u8bdd\u6982\u62ec：\u7b97\u672a\u77e5\u6570 x0 \u7528\u5c0f\u6a21\u6570 (mod)，\u7b97\u5927\u7b54\u6848 ans \u7528\u5927\u6a21\u6570 (lcm)\u3002<\/p>\n

          \u5176\u4ed6\u6613\u9519\u70b9：<\/h5>\n
            \n
          • \n

            \u591a\u7ec4\u6570\u636e\u4e0e EOF \u8bfb\u5165：\u9898\u76ee\u660e\u786e\u8bf4\u660e\u201c\u591a\u7ec4\u6570\u636e\u201d，\u5fc5\u987b\u5199 while(cin >> n)\u3002<\/p>\n<\/li>\n

          • \n

            \u8f93\u5165\u987a\u5e8f\u9677\u9631：POJ 2891 \u7684\u8f93\u5165\u683c\u5f0f\u662f\u5148\u7ed9\u6a21\u6570 m[i]\u200b，\u518d\u7ed9\u4f59\u6570 a[i]\u200b\u3002\u76f4\u63a5\u8bfb\u9519\u4f4d\u4f1a\u5bfc\u81f4\u5e95\u5c42\u6269\u6b27\u5f7b\u5e95\u9519\u8bef\u3002<\/p>\n<\/li>\n

          • \n

            \u6ea2\u51fa\u4e0e\u00a0__int128：\u5927\u6574\u6570\u76f8\u4e58\u5982 x0*lcm \u5728\u6781\u7aef\u6570\u636e\u4e0b\u53ef\u80fd\u4f1a\u6491\u7206long long，\u5efa\u8bae\u5957\u4e0a __int128\u00a0，\u4f46\u662f\u4e0d\u5199\u4e5f\u80fd\u8fc7\u3002<\/p>\n<\/li>\n<\/ul>\n

            6. \u5b8c\u6574\u4ee3\u7801<\/h4>\n

            \/\/\u56e0\u4e3a\u4e0d\u786e\u5b9a\u6a21\u6570\u662f\u5426\u4e92\u8d28 \u6240\u4ee5\u4f7f\u7528\u6269\u5c55\u4e2d\u56fd\u5269\u4f59\u5b9a\u7406
            \n#include <iostream>
            \nusing namespace std;
            \nint n;
            \nlong long a[100010];\/\/a[i]\u4e0em[i]\u5bf9\u5e94，\u4ee3\u8868\u7b2cm\u4e2a\u5f0f\u5b50\u7684\u6a21\u6570\u60c5\u51b5\u4e0b\u6240\u5bf9\u5e94\u4f59\u6570
            \nlong long m[100010];\/\/m[i]\u4ee3\u8868\u7b2ci\u4e2a\u5f0f\u5b50\u7684\u6a21\u6570
            \ntypedef long long ll;<\/p>\n

            \/\/\u6269\u5c55\u6b27\u51e0\u91cc\u5f97
            \n\/\/\u6c42ax+by=gcd(a,b)\u7684\u7279\u89e3 \u5e76\u8fd4\u56de\u6700\u5927\u516c\u7ea6\u6570d
            \nll exgcd(ll a,ll b,ll &x_,ll &y_){
            \n if(b==0){
            \n x_=1;
            \n y_=0;
            \n return a;
            \n }
            \n ll d=exgcd(b,a%b,x_,y_);
            \n ll tmp=x_;
            \n x_=y_;
            \n y_=tmp-(a\/b)*y_;
            \n return d;
            \n}<\/p>\n

            \/\/\u6269\u5c55\u4e2d\u56fd\u5269\u4f59\u5b9a\u7406
            \nll excrt(){
            \n ll ans=a[1];\/\/\u6ee1\u8db3\u7b2c\u4e00\u4e2a\u5f0f\u5b50\u7684\u7ed3\u679c（\u89e3）
            \n \/\/lcm\u4ee3\u8868\u5f53\u524d\u6240\u6709\u5408\u5e76\u65b9\u7a0b\u7684\u6700\u5c0f\u516c\u500d\u6570 \u8fd9\u91cc\u662f\u7b2c\u4e00\u4e2a\u65b9\u7a0b\u7684\u6a21\u6570
            \n ll lcm=m[1];<\/p>\n

            \/\/\u6bcf\u4e00\u8f6e\u7684\u65b9\u7a0b\u90fd\u8981\u6ee1\u8db3
            \n for(int i=2;i<=n;i++){
            \n ll x0,y0;
            \n \/\/\u8981\u524d\u4e00\u8f6e\u7684\u5f0f\u5b50\u6ee1\u8db3\u540e\u4e00\u8f6e\u7684\u8981\u6c42 \u5219\u4ee3\u8868\u7b2ci\u8f6e\u7684\u76ee\u6807\u65b9\u7a0b\u4e3a
            \n \/\/ans+x0*lcm\u2261a[i] (mod m[i]) （x0\u662f\u8fd9\u4e00\u8f6e\u65b0\u8bbe\u7684\u672a\u77e5\u91cf \u4ee3\u8868\u52a0x0\u4e2alcm\u53ef\u4ee5\u6ee1\u8db3\u8981\u6c42）
            \n \/\/\u79fb\u9879 lcm*x0+m[i]*y0=a[i]-ans (y0\u662f\u8fd9\u4e00\u8f6e\u65b0\u8bbe\u7684\u672a\u77e5\u91cf)
            \n \/\/\u521b\u5efa\u4e00\u4e2a\u53d8\u91cfc=a[i]-ans
            \n ll c=a[i]-ans;
            \n \/\/c\u6709\u53ef\u80fd\u4e3a\u8d1f\u6570 \u628ac\u53d6\u6b63
            \n c=(c%m[i]+m[i])%m[i];
            \n \/\/\u6269\u5c55\u6b27\u51e0\u91cc\u5f97\u6c42\u89e3d\u548cx0 lcm*x0+m[i]*y0=d
            \n ll d=exgcd(lcm,m[i],x0,y0);<\/p>\n

            \/\/\u6839\u636e\u88f4\u8700\u5b9a\u7406 \u5982\u679cc\u4e0d\u80fd\u88abd\u6574\u9664，\u5219\u65e0\u89e3
            \n if(c%d!=0) return -1;<\/p>\n

            \/\/\u7b97\u51fa\u8981\u6269\u5927\u7684\u500d\u6570 \u56e0\u4e3aexgcd\u6c42\u7684\u662flcm*x0+m[i]*y0=d\u7684\u60c5\u51b5
            \n \/\/\u6c42lcm*x0+m[i]*y0=c x0 y0\u80af\u5b9a\u8981\u76f8\u5e94\u6269\u5927
            \n ll times=c\/d;<\/p>\n

            \/\/\u6a21\u6570\u8981\u76f8\u5e94\u53d8\u5316 \u7b49\u5f0f\u4e24\u8fb9\u540c\u9664\u4ee5d\u540e \u5c40\u90e8\u8fd0\u7b97\u7684\u5468\u671f\u53d8\u4e3am[i]\/d
            \n ll mod=m[i]\/d;<\/p>\n

            \/\/\u628ax0\u4ece\u53ef\u80fd\u7684\u8d1f\u6570\u5148\u8f6c\u5316\u6210\u767e\u5206\u767e\u7684\u6b63\u6570
            \n x0=(x0%mod+mod)%mod;<\/p>\n

            \/\/\u7b49\u6bd4\u4f8b\u6269\u5927x0
            \n x0=(x0*times)%mod;<\/p>\n

            \/\/\u4e0b\u4e00\u8f6e\u603b\u5468\u671f
            \n ll nextlcm=lcm\/d*m[i];<\/p>\n

            \/\/\u8fd9\u4e00\u8f6e\u7684\u89e3
            \n \/\/\u66f4\u65b0\u5168\u5c40\u7b54\u6848=\u65e7\u7b54\u6848+\u8de8\u8d8a\u7684\u6b65\u6570*\u65e7\u5468\u671f
            \n \/\/\u66f4\u65b0\u5b8c\u6bd5\u540e，\u5fc5\u987b\u5bf9\u65b0\u7684\u5168\u5c40\u5468\u671fnextlcm\u53d6\u6a21
            \n ans=(ans+x0*lcm)%nextlcm;<\/p>\n

            \/\/\u66f4\u65b0\u603b\u5468\u671f
            \n lcm=nextlcm;<\/p>\n

            \/\/\u786e\u4fdd\u603b\u7b54\u6848\u662f\u6700\u5c0f\u6b63\u6574\u6570
            \n ans=(ans%lcm+lcm)%lcm;
            \n }
            \n return ans;
            \n}<\/p>\n

            int main(){
            \n ios::sync_with_stdio(false);
            \n cin.tie(0);
            \n while(cin>>n){
            \n for(int i=1;i<=n;i++)
            \n cin>>m[i]>>a[i];<\/p>\n

            \/\/\u6269\u5c55\u4e2d\u56fd\u5269\u4f59\u5b9a\u7406\u6c42\u89e3
            \n cout<<excrt()<<"\\\\n";
            \n }
            \n return 0;<\/p>\n

            }<\/p>\n","protected":false},"excerpt":{"rendered":"

            \u3010\u9898\u76ee\u63cf\u8ff0\u3011\u539f\u9898\u6765\u81ea：POJ 2891\u7ed9\u5b9a\u00a02n\u00a0\u4e2a\u6b63\u6574\u6570\u00a0a1,a2,\u22ef,an\u548c\u00a0m1,m2,\u22ef,m[n] ，\u6c42\u4e00\u4e2a\u6700\u5c0f\u7684\u6b63\u6574\u6570\u00a0x，\u6ee1\u8db3\u00a0\u2200i\u2208[1,n],x\u2261ai(mod m[i])，\u6216\u8005\u7ed9\u51fa\u65e0\u89e3\u3002\u3010\u8f93\u5165\u3011\u591a\u7ec4\u6570\u636e\u3002\u6bcf\u7ec4\u6570\u636e\u7b2c\u4e00\u884c\u4e00\u4e2a\u6574\u6570\u00a0n；\u63a5\u4e0b\u6765\u00a0n\u00a0\u884c，\u6bcf\u884c\u4e24\u4e2a\u6574\u6570\u00a0mi,ai\u00a0\u3002\u3010\u8f93\u51fa\u3011\u5bf9\u4e8e\u6bcf\u7ec4\u6570\u636e，\u82e5\u65e0\u89e3，\u8f93\u51fa\u00a0\u22121；\u5426\u5219\u8f93\u51fa\u4e00\u4e2a\u975e\u8d1f\u6574<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[12969,12970,12971,4145,427],"topic":[],"class_list":["post-113998","post","type-post","status-publish","format-standard","hentry","category-server","tag-12969","tag-12970","tag-12971","tag-4145","tag-427"],"yoast_head":"\n\u3010\u4f8b 5\u3011Strange Way to Express Integers\uff08\u4fe1\u606f\u5b66\u5965\u8d5b\u4e00\u672c\u901a- P1635\uff09 - 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