Happy 2004（快速冪+乘法逆元）

標籤：

Happy 2004

問題描述 :

Consider a positive integer X,and let S be the sum of all positive integer divisors of 2004^X. Your job is to determine S modulo 29 (the rest of the division of S by 29).

 

Take X = 1 for an example. The positive integer divisors of 2004^1 are 1, 2, 3, 4, 6, 12, 167, 334, 501, 668, 1002 and 2004. Therefore S = 4704 and S modulo 29 is equal to 6.

輸入:

The input consists of several test cases. Each test case contains a line with the integer X (1 <= X <= 10000000).

 

A test case of X = 0 indicates the end of input, and should not be processed.

輸出:

For each test case, in a separate line, please output the result of S modulo 29.

範例輸入:

1100000

範例輸出:

610

 

設S(x)表示x的因子和。則題目求為:S(2004^X)mod 29
因子和S是積性函數,即滿足性質1。

性質1 ：如果 gcd(a,b)=1  則 S(a*b)= S(a)*S(b)
2004^X=4^X * 3^X *167^X
S(2004^X)=S(2^(2X)) * S(3^X) * S(167^X)

性質2 ：如果 p 是素數 則 S(p^X)=1+p+p^2+…+p^X = (p^(X+1)-1)/(p-1)
因此：S(2004^X)=(2^(2X+1)-1) * (3^(X+1)-1)/2 * (167^(X+1)-1)/166
167%29 == 22
S(2004^X)=(2^(2X+1)-1) * (3^(X+1)-1)/2 * (22^(X+1)-1)/21

性質3 ：(a*b)/c %M= a%M * b%M * inv(c)
其中inv(c)即滿足 (c*inv(c))%M=1的最小整數,這裡M=29
則inv(1)=1，inv(2)=15，inv(22)=15

有上得：
S(2004^X)=(2^(2X+1)-1) * (3^(X+1)-1)/2 * (22^(X+1)-1)/21
=(2^(2X+1)-1) * (3^(X+1)-1)*15 * (22^(X+1)-1)*18

 1 #include <cstdio> 2 #include <iostream> 3 #include <algorithm> 4 #include <cstring> 5 #include <cmath> 6 #include <queue> 7 #include <map> 8 #include <set> 9 using namespace std;10 #define LL long long11 const int INF=0x3f3f3f3f;12 const double eps=1e-5;13 int p=29;14 LL pow_mod(LL x,LL n)15 {16     LL res=1;17     while(n>0)18     {19         if(n&1)    res=res*x%p;20         x=x*x%p;21         n>>=1;22     }23     return res;24 }25 int main()26 {27     LL x,i;28     while(cin>>x&&x)29     {30         int a=pow_mod(2,2*x+1);31         int b=pow_mod(3,x+1);32         int c=pow_mod(22,x+1);33         int s=((a-1)*(b-1)*(c-1)*15*18)%29;34         cout<<s<<endl;35      }36 37 }

 

Happy 2004（快速冪+乘法逆元）