codeforces magic five --快速幂模

题目链接:http://codeforces.com/contest/327/problem/C

首先先算出一个周期里面的值,保存在ans里面,就是平常的快速幂模m做法.

然后要计算一个公式,比如有k个部分,那么对于没一个位置i, 都有2^i + 2^(i+n) + ... + 2^(i+(k-1)*n) = 2^i(1 + 2^n + ... + 2^((k-1)*n)) = 2^i * (1-2^(n*k))/(1-2^n)

所以结果就是ans * (1-2^(n*k))/(1-2^n) % MOD;

然后就是关键计算(1-2^(n*k))/(1-2^n) % MOD;

用到费马小定理a^(p-1)同余于1(mod 1).p是一个质数,那么a^(p-2) * a 同余于1(mod 1) ,所以a  的逆元就是 a^(p-2)

MOD是一个质数,所以(1-2^(n*k))/(1-2^n) % MOD = (2^(n*k)-1)/(2^n-1) % MOD = (2^(n*k)-1)%MOD * ((2^n-1)^(MOD-2))%MOD

 1 /*
 2  * =====================================================================================
 3  *       Filename:  magic.cpp
 4  *        Created:  19/07/2013 12:27:18
 5  *         Author:  liuxueyang (lxy), 1459917536@qq.com
 6  *   Organization:  Hunan University
 7  *
 8  * =====================================================================================
 9  */
10 
11 /*
12 ID: zypz4571
13 LANG: C++
14 TASK: magic
15 */
16 #include <cstdlib>
17 #include <iostream>
18 #include <cstdio>
19 #include <cstring>
20 #include <cmath>
21 #include <cctype>
22 #include <algorithm>
23 #include <queue>
24 #include <set>
25 #include <stack>
26 #include <map>
27 #include <list>
28 using namespace std;
29 #define INF 0x3f3f3f3f
30 const double eps=1e-9;
31 char a[100010];
32 const int MOD=1000000007;
33 #define LL long long
34 int k;
35 LL quick(LL a, LL b) {
36     LL ans=1;
37     while (b) {
38         if(b&1) ans=(ans*a)%MOD; b/=2; a*=a; a%=MOD;
39     }
40     return ans;
41 }
42 int main ( int argc, char *argv[] )
43 {
44 #ifndef ONLINE_JUDGE
45     freopen("in.txt", "r", stdin);
46 #endif
47     int k; string s; cin>>s>>k; LL n=s.size();
48     LL a=quick(2,n*k)-1+MOD; a=(a+MOD)%MOD;
49     LL b=quick((quick(2,n)-1+MOD)%MOD,MOD-2); b=(b+MOD)%MOD;
50     LL ans=0;
51     for(size_t i = 0; i < n; ++i) if (s[i]=='0'||s[i]=='5') ans=(ans+quick(2,i))%MOD;
52     cout<<a*b%MOD*ans%MOD<<endl;
53     return EXIT_SUCCESS;
54 }                /* ----------  end of function main  ---------- */

搞定,收工

posted on 2013-07-19 16:29  aries__liu  阅读(285)  评论(0编辑  收藏  举报