{"id":44389,"date":"2025-06-18T17:17:46","date_gmt":"2025-06-18T09:17:46","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/44389.html"},"modified":"2025-06-18T17:17:46","modified_gmt":"2025-06-18T09:17:46","slug":"%e3%80%90%e7%a7%a6%e4%b9%9d%e7%bb%8d%e7%ae%97%e6%b3%95%e3%80%91%e5%b0%8f%e7%ba%a2%e7%9a%84-gcd","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/44389.html","title":{"rendered":"\u3010\u79e6\u4e5d\u7ecd\u7b97\u6cd5\u3011\u5c0f\u7ea2\u7684 gcd"},"content":{"rendered":"

\u9898\u76ee<\/h2>\n

\u725b\u5ba2\u7f51：\u5c0f\u7ea2\u7684 gcd \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"<\/p>\n

\u9898\u76ee\u5206\u6790<\/h2>\n

\u6211\u4eec\u77e5\u9053，\u6c42gcd\u5c31\u7528\u6b27\u51e0\u91cc\u5f97\u7b97\u6cd5（\u8f97\u8f6c\u76f8\u9664\u6cd5）：gcd(a,b)=gcd(b,a mod b)\u3002\u4f46\u662f\u8fd9\u9898\u7684a\u975e\u5e38\u5927，\u6700\u5927\u662f\u4e00\u4e2a1e6\u4f4d\u6570，\u65e0\u6cd5\u4f7f\u7528\u4efb\u4f55\u6570\u636e\u7c7b\u578b\u5b58\u50a8\u3002\u5982\u679c\u4f7f\u7528\u9ad8\u7cbe\u5ea6\u8ba1\u7b97\u7684\u8bdd，\u9700\u8981\u9010\u4f4d\u8ba1\u7b97，\u4f1a\u8d85\u65f6\u3002<\/p>\n

\u4e00\u4e2a\u5341\u8fdb\u5236\u6570\u5b57\u4f8b\u5982 <\/p>\n

123 <\/p>\n

123 <\/p>\n

<\/span><\/span>123<\/span><\/span><\/span><\/span><\/span> \u662f\u53ef\u4ee5\u62c6\u5206\u6210 <\/p>\n

1 <\/p>\n

\u2217 <\/p>\n

10 <\/p>\n

2 <\/p>\n

+ <\/p>\n

2 <\/p>\n

\u2217 <\/p>\n

10 <\/p>\n

1 <\/p>\n

+ <\/p>\n

3 <\/p>\n

\u2217 <\/p>\n

10 <\/p>\n

0 <\/p>\n

1*10^2 + 2*10^1 + 3*10^0 <\/p>\n

<\/span><\/span>1<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>1<\/span>0<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>2<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>1<\/span>0<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>3<\/span><\/span>\u2217<\/span><\/span><\/span><\/span>1<\/span>0<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u800c\u4e14\u6b27\u51e0\u91cc\u5f97\u7b97\u6cd5\u4f1a\u4e0d\u65ad\u5bf9a\u53d6\u6a21，\u6211\u4eec\u53c8\u77e5\u9053\u53ef\u4ee5\u5728\u4efb\u610f\u4f4d\u7f6e\u53d6\u6a21\u7684\u6027\u8d28\u3002\u6839\u636e\u8fd9\u4e24\u4e2a\u7279\u6027，\u6211\u4eec\u5c31\u53ef\u4ee5\u901a\u8fc7\u79e6\u4e5d\u7ecd\u7b97\u6cd5\u5c06 a \u8fd9\u4e2a\u6570\u5b57\u62c6\u520610\u7684\u4e00\u5143\u591a\u9879\u5f0f，\u5728\u9010\u4f4d\u8ba1\u7b97a\u7684\u540c\u65f6\u8ba1\u7b97a mod b\u3002<\/p>\n

<\/p>\n

a
\n \u200a <\/p>\n

m <\/p>\n

o <\/p>\n

d <\/p>\n

\u200a <\/p>\n

b <\/p>\n

= <\/p>\n

( <\/p>\n

\u2211 <\/p>\n

i <\/p>\n

= <\/p>\n

0 <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

1 <\/p>\n

d <\/p>\n

i <\/p>\n

\u00d7 <\/p>\n

10 <\/p>\n

i <\/p>\n

) <\/p>\n

\u200a <\/p>\n

m <\/p>\n

o <\/p>\n

d <\/p>\n

\u200a <\/p>\n

b <\/p>\n

a \\\\bmod b = \\\\left( \\\\sum_{i=0}^{n-1} d_i \\\\times 10^i \\\\right) \\\\bmod b <\/p>\n

<\/span><\/span>a<\/span><\/span><\/span>mod<\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span><\/span><\/span>i<\/span>=<\/span>0<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>n<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>1<\/span>0<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>mod<\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span><\/span> \u5176\u4e2d <\/p>\n

d <\/p>\n

i <\/p>\n

d_i <\/p>\n

<\/span><\/span>d<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f <\/p>\n

a <\/p>\n

a <\/p>\n

<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span> \u7684\u7b2c <\/p>\n

i <\/p>\n

i <\/p>\n

<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span> \u4f4d\u6570\u5b57\u3002<\/p>\n

\u79e6\u4e5d\u7ecd\u7b97\u6cd5<\/h3>\n

\u662f\u4e00\u79cd\u591a\u9879\u5f0f\u6c42\u503c\u7684\u9ad8\u6548\u7b97\u6cd5\u3002\u5b83\u7684\u6838\u5fc3\u601d\u60f3\u662f\u901a\u8fc7\u5d4c\u5957\u4e58\u6cd5\u548c\u52a0\u6cd5\u6765\u51cf\u5c11\u8ba1\u7b97\u6b21\u6570，\u5c06\u591a\u9879\u5f0f\u6c42\u503c\u7684\u65f6\u95f4\u590d\u6742\u5ea6 <\/p>\n

O <\/p>\n

( <\/p>\n

n <\/p>\n

2 <\/p>\n

) <\/p>\n

O(n^2) <\/p>\n

<\/span><\/span>O<\/span>(<\/span>n<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u4f18\u5316\u5230 <\/p>\n

O <\/p>\n

( <\/p>\n

n <\/p>\n

) <\/p>\n

O(n) <\/p>\n

<\/span><\/span>O<\/span>(<\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u5bf9\u4e8e\u4e00\u4e2a\u591a\u9879\u5f0f： <\/p>\n

f <\/p>\n

( <\/p>\n

x <\/p>\n

) <\/p>\n

= <\/p>\n

a <\/p>\n

n <\/p>\n

x <\/p>\n

n <\/p>\n

+ <\/p>\n

a <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

1 <\/p>\n

x <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

1 <\/p>\n

+ <\/p>\n

a <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

2 <\/p>\n

x <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

2 <\/p>\n

+ <\/p>\n

\u22ef <\/p>\n

+ <\/p>\n

a <\/p>\n

1 <\/p>\n

x <\/p>\n

1 <\/p>\n

+ <\/p>\n

a <\/p>\n

0 <\/p>\n

x <\/p>\n

0 <\/p>\n

f(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \\\\dots + a_1 x^1 + a_0 x^0 <\/p>\n

<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>a<\/span><\/span>n<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>n<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>a<\/span><\/span>n<\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>n<\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span>+<\/span><\/span><\/span><\/span>a<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>a<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u79e6\u4e5d\u97f6\u7b97\u6cd5\u5c06\u5176\u6539\u5199\u4e3a\u5d4c\u5957\u5f62\u5f0f：<\/p>\n

<\/p>\n

f <\/p>\n

( <\/p>\n

x <\/p>\n

) <\/p>\n

= <\/p>\n

( <\/p>\n

a <\/p>\n

n <\/p>\n

x <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

1 <\/p>\n

+ <\/p>\n

a <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

1 <\/p>\n

x <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

2 <\/p>\n

+ <\/p>\n

a <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

2 <\/p>\n

x <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

3 <\/p>\n

+ <\/p>\n

\u22ef <\/p>\n

+ <\/p>\n

a <\/p>\n

1 <\/p>\n

) <\/p>\n

x <\/p>\n

+ <\/p>\n

a <\/p>\n

0 <\/p>\n

= <\/p>\n

( <\/p>\n

( <\/p>\n

a <\/p>\n

n <\/p>\n

x <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

2 <\/p>\n

+ <\/p>\n

a <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

1 <\/p>\n

x <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

3 <\/p>\n

+ <\/p>\n

a <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

2 <\/p>\n

x <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

4 <\/p>\n

+ <\/p>\n

\u22ef <\/p>\n

+ <\/p>\n

a <\/p>\n

2 <\/p>\n

) <\/p>\n

x <\/p>\n

+ <\/p>\n

a <\/p>\n

1 <\/p>\n

) <\/p>\n

x <\/p>\n

+ <\/p>\n

a <\/p>\n

0 <\/p>\n

\u00a0
\n \u2009 <\/p>\n

\u22ee <\/p>\n

= <\/p>\n

( <\/p>\n

\u2026 <\/p>\n

( <\/p>\n

( <\/p>\n

a <\/p>\n

n <\/p>\n

x <\/p>\n

+ <\/p>\n

a <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

1 <\/p>\n

) <\/p>\n

x <\/p>\n

+ <\/p>\n

a <\/p>\n

n <\/p>\n

\u2212 <\/p>\n

2 <\/p>\n

) <\/p>\n

x <\/p>\n

+ <\/p>\n

\u22ef <\/p>\n

+ <\/p>\n

a <\/p>\n

2 <\/p>\n

) <\/p>\n

x <\/p>\n

+ <\/p>\n

a <\/p>\n

1 <\/p>\n

) <\/p>\n

x <\/p>\n

+ <\/p>\n

a <\/p>\n

0 <\/p>\n

\\\\begin{aligned} f(x) &= (a_n x^{n-1} + a_{n-1} x^{n-2} + a_{n-2} x^{n-3} + \\\\dots + a_1 )x + a_0 \\\\\\\\ &= \\\\left( (a_n x^{n-2} + a_{n-1} x^{n-3} + a_{n-2} x^{n-4} + \\\\dots + a_2 )x + a_1 \\\\right)x + a_0 \\\\\\\\ &\\\\ \\\\, \\\\vdots \\\\\\\\ &= \\\\left( \\\\dots \\\\left( (a_n x + a_{n-1} )x + a_{n-2} \\\\right)x + \\\\dots + a_2 \\\\right)x + a_1 )x + a_0 \\\\end{aligned} <\/p>\n

<\/span><\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>(<\/span>a<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>n<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>a<\/span><\/span>n<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>n<\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>a<\/span><\/span>n<\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>n<\/span>\u2212<\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>\u22ef<\/span><\/span>+<\/span><\/span>a<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>x<\/span><\/span>+<\/span><\/span>a<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>(<\/span><\/span>(<\/span>a<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>n<\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>a<\/span><\/span>n<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>n<\/span>\u2212<\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>a<\/span><\/span>n<\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>n<\/span>\u2212<\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>\u22ef<\/span><\/span>+<\/span><\/span>a<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>x<\/span><\/span>+<\/span><\/span>a<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>x<\/span><\/span>+<\/span><\/span>a<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0<\/span><\/span>\u22ee<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>(<\/span>\u2026<\/span><\/span>(<\/span>(<\/span>a<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>+<\/span><\/span>a<\/span><\/span>n<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>x<\/span><\/span>+<\/span><\/span>a<\/span><\/span>n<\/span>\u2212<\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span>x<\/span><\/span>+<\/span><\/span>\u22ef<\/span><\/span>+<\/span><\/span>a<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span>x<\/span><\/span>+<\/span><\/span>a<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>x<\/span><\/span>+<\/span><\/span>a<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u793a\u4f8b： \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"<\/p>\n

AC\u4ee3\u7801<\/h2>\n

#<\/span>include<\/span><iostream><\/span> <\/span><\/p>\n

using<\/span> namespace<\/span> std;<\/span><\/p>\n

string a;<\/span>
\nint<\/span> b;<\/span><\/p>\n

int<\/span> calc<\/span>(<\/span>)<\/span>
\n{<\/span>
\nlong<\/span> long<\/span> ret =<\/span> 0<\/span>;<\/span>
\nfor<\/span>(<\/span>auto<\/span> ch:<\/span>a)<\/span>
\n{<\/span>
\nret =<\/span> (<\/span>ret *<\/span> 10<\/span> +<\/span> ch –<\/span> '0'<\/span>)<\/span> %<\/span> b;<\/span>
\n}<\/span>
\nreturn<\/span> ret;<\/span>
\n}<\/span><\/p>\n

int<\/span> gcd<\/span>(<\/span>int<\/span> a,<\/span> int<\/span> b)<\/span>
\n{<\/span>
\nreturn<\/span> b ==<\/span> 0<\/span> ?<\/span> a :<\/span> gcd<\/span>(<\/span>b,<\/span>a%<\/span>b)<\/span>;<\/span>
\n}<\/span><\/p>\n

int<\/span> main<\/span>(<\/span>)<\/span>
\n{<\/span>
\ncin >><\/span> a >><\/span> b;<\/span><\/p>\n

cout <<<\/span> gcd<\/span>(<\/span>calc<\/span>(<\/span>)<\/span>,<\/span>b)<\/span> <<<\/span> endl;<\/span><\/p>\n

return<\/span> 0<\/span>;<\/span>
\n}<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

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