生理周期

http://poj.org/problem?id=1006

中国剩余定理模板题

 1 #include<stdio.h>

 2 #include<string.h>

 3 int p,e,i,d;

 4 int a[5],m[5];

 5 int item = 1;

 6 int extend_gcd(int a,int b,int &x,int &y)

 7 {

 8     if(b == 0)

 9     {

10         x = 1;

11         y = 0;

12         return a;

13     }

14     else

15     {

16         int r = extend_gcd(b,a%b,x,y);

17         int t = x;

18         x = y;

19         y = t-a/b*y;

20         return r;

21     }

22 }

23 

24 void CRT()

25 {

26     int M = 1,tm,ret = 0;

27     for(int i = 0; i < 3; i++)

28         M*=m[i];

29     for(int i = 0; i < 3; i++)

30     {

31         int x,y;

32         tm = M/m[i];

33         extend_gcd(tm,m[i],x,y);

34         ret = (ret + tm*x*a[i])%M;

35     }

36     if(ret - d <= 0)

37         printf("Case %d: the next triple peak occurs in %d days.\n",item,ret+M-d);

38     else printf("Case %d: the next triple peak occurs in %d days.\n",item,(ret+M)%M-d);

39 }

40 

41 int main()

42 {

43 

44     while(~scanf("%d %d %d %d",&p,&e,&i,&d))

45     {

46         if(p == -1 && e == -1 && i == -1 && d == -1)

47             break;

48         a[0] = p;

49         a[1] = e;

50         a[2] = i;

51         m[0] = 23;

52         m[1] = 28;

53         m[2] = 33;

54         CRT();

55         item++;

56 

57     }

58     return 0;

59 }

View Code

中国剩余定理

设m1,m2,...,mk两两互质,则下面的同余方程组：
  x = a1 (mod m1)
  x = ak (mod mk)
    .
    .
    .
  x = ak (mod mk)

在0<=x<M = m1*m2*...*mk内有唯一解。
记Mi = M / mi (1<=i<=k) ,因为(Mi,mi)=1,故有两整数pi，qi满足Mi*pi + mi*qi = 1，
如果记ei = Mi*pi ，那么有 
ei = 0 (mod mj) , j<>i
   = 1 (mod mj) , j==i
   
很显然，e1*a1 + e2*a2 + ... + ek*ak 就时方程组的一个解 ，
这个解%M后就得到了最小的非负整数解；

求ei可以用扩展欧几里得来求；