C - (例题)整数分解,计数
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Description
Given two positive integers G and L, could you tell me how many solutions of (x, y, z) there are, satisfying that gcd(x, y, z) = G and lcm(x, y, z) = L?
Note, gcd(x, y, z) means the greatest common divisor of x, y and z, while lcm(x, y, z) means the least common multiple of x, y and z.
Note 2, (1, 2, 3) and (1, 3, 2) are two different solutions.
Input
First line comes an integer T (T <= 12), telling the number of test cases.
The next T lines, each contains two positive 32-bit signed integers, G and L.
It’s guaranteed that each answer will fit in a 32-bit signed integer.
Output
For each test case, print one line with the number of solutions satisfying the conditions above.
Sample Input
2
6 72
7 33
Sample Output
72
0
题目大意:
给你两个数L,G,问你有多少有序数组(x,y,z)满足GCD(x,y,z)=G,LCM(x,y,z)=L,首先如果gcd(x,y,z)=G,
那么有gcd(x/G,y/G,z/G)=1(说明这三个数两两互素),此时应该满足lcm(x,y,z)=L/G,要求L/G为整数,则若L%G==0,则一定有解,(x,y,z都等于L/G即可)
反之无解
此时将L/G作正整数唯一分解,T=L/G=a1^b1*a2^b2*.......*an^bn,对于a1,要满足gcd(x/g,y/g,z/g)=1,a1^k则至少有一个k=0,同时
还得满足lcm(x/g,y/g,z/g)=l/g,则至少有一个k=b1,这样就有三种情况(0,0,b1)(b1,b1,0)(0,1~b1-1,b1)共有6+6(b1-1)=6*b1种,其他的同理
由分步乘法计数原理,最终答案为(6*b1)*(6*b2)*........(6*bn)
代码:
#include <iostream> #include <cstdio> #include <cstring> #include<algorithm> #include <cmath> using namespace std; typedef long long ll; const int maxn=2e5;// bool vis[maxn]; ll prime[maxn/10]; int tot; void getprime()//因为n的范围是1e14,打表只需要打到sqrt(n)即可,最多只可能有一个素因子大于sqrt(n),最后特判一下即可; { memset(vis,true,sizeof(vis)); tot=0; for(ll i=2;i<maxn;i++) { if(vis[i]) { prime[tot++]=i; for(ll j=i*i;j<maxn;j+=i) { vis[j]=false; } } } } /*void Eulerprime() { memset(vis,true,sizeof(vis)); int tot=0; for(int i=2;i<maxn;i++) { if(vis[i]) prime[tot++]=i; for(int j=0;j<tot&&prime[j]*i<maxn;j++) { vis[i*prime[j]]=false; if(i%prime[j]==0) break; } } }*/ int a[1000],b[1000]; int cnt=0; void sbreak(ll n)//正整数唯一分解 { memset(a,0,sizeof(a)); memset(b,0,sizeof(b)); cnt=0; for(int i=0;prime[i]*prime[i]<=n;i++) { if(n%prime[i]==0) { a[cnt]=prime[i]; while(n%prime[i]==0) { b[cnt]++; n/=prime[i]; } cnt++; } } if(n!=1) { a[cnt]=n; b[cnt]=1; cnt++;//为了使两种情况分解后素因子下标都是0~cnt-1; } } int pow_mod(int m,int n) { ll pw=1; while(n) { if(n&1) pw*=m; m*=m; n/=2; } return pw; } int kase; int main() { int T; ll L,G; getprime(); scanf("%d",&T); kase=0; while(T--) { scanf("%lld%lld",&G,&L); if(L%G) {printf("0\n");continue;} ll n=L/G; sbreak(n); ll sum=1; for(int i=0;i<cnt;i++) { sum*=(6*b[i]); } printf("%lld\n",sum); } }