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  • hdu2669-Romantic-(扩展欧几里得定理)

    Romantic

    Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 32768/32768 K (Java/Others)
    Total Submission(s): 10883    Accepted Submission(s): 4610


    Problem Description
    The Sky is Sprite.
    The Birds is Fly in the Sky.
    The Wind is Wonderful.
    Blew Throw the Trees
    Trees are Shaking, Leaves are Falling.
    Lovers Walk passing, and so are You.
    ................................Write in English class by yifenfei



    Girls are clever and bright. In HDU every girl like math. Every girl like to solve math problem!
    Now tell you two nonnegative integer a and b. Find the nonnegative integer X and integer Y to satisfy X*a + Y*b = 1. If no such answer print "sorry" instead.
     
    Input
    The input contains multiple test cases.
    Each case two nonnegative integer a,b (0<a, b<=2^31)
     
    Output
    output nonnegative integer X and integer Y, if there are more answers than the X smaller one will be choosed. If no answer put "sorry" instead.
     
    Sample Input
    77 51
    10 44
    34 79
     
    Sample Output
    2 -3
    sorry
    7 -3
     
    翻译:输入a和b,找到x和y满足ax+by=1,x为正数。
    分析:显然是扩展欧几里得定理的模板题,当gcd(a,b)=1时,有解,然而扩欧模板解出来是特解,不能保证x为正数,需要遍历通解。通解公式:x = x0 + (b/gcd)*t; y = y0 + (a/gcd)*t;
    注意:为什么是b/gcd和a/gcd,而不是b和a?
    a(x+b*t/gcd) + b(y-a*t/gcd) = gcd;
    ax + a*b*t/gcd + by - b*a*t/gcd = gcd = ax + by;
    对于x = x0 + b*t和y = y0 - a*t;
    a(x+bt) + b(y-at) = gcd
    ax+abt + by-abt = gcd = ax + by
    显然b/gcd比b更小,通解找得全面一些。
     1 #include<stdio.h>
     2 #include<iostream>
     3 #include<algorithm>
     4 #include<cstring>
     5 #include<math.h>
     6 #include<string>
     7 #define ll long long
     8 #define inf 0x3f3f3f3f
     9 using namespace std;
    10 // ax + by = gcd(a,b)
    11 ll exgcd(ll a, ll b, ll &x, ll &y)//扩展欧几里德定理
    12 {
    13     if(b==0)//终有一次a%b传进来是0,递归出口
    14     {
    15         x=1;y=0;
    16         return a;
    17     }
    18     ll q=exgcd(b,a%b,y,x);
    19     //最终递归出来,y1=1,x1=0
    20     y=y-(a/b)*x;
    21     //后面的y相当于下一个递归的x2,x相当于下一个递归的y2,符合推导公式
    22     //x1=y2;   y1=x2-[a/b]*y2;
    23     return q;
    24 }
    25 
    26 int main()
    27 {
    28     ll a,b;
    29     while(scanf("%lld %lld",&a,&b)!=EOF)
    30     {
    31         ll x,y;
    32         ll gcd=exgcd(a,b,x,y);
    33         if(gcd==1)
    34         {
    35             while(x<0)
    36             {
    37                 x=x+b;
    38                 y=y-a;
    39             }
    40             printf("%lld %lld
    ",x,y);
    41         }
    42         else printf("sorry
    ");
    43     }
    44     return 0;
    45 }
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  • 原文地址:https://www.cnblogs.com/shoulinniao/p/10359297.html
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