HDU 4497 GCD and LCM (素数筛选+算术基本定理)

算术基本定理可表述为：任何一个大于1的自然数 N,如果N不为质数，那么N可以唯一分解成有限个质数的乘积N=P1a1P2a2P3a3......Pnan，这里P1<P2<P3......<Pn均为质数，其中指数ai是正整数。这样的分解称为 N 的标准分解式。最早证明是由欧几里得给出的，现代是由陈述证明。此定理可推广至更一般的交换代数和代数数论。

lcm(x,y,z)=k;

gcd(x,y,z)=t;

若：x=a*t; y=b*t; z=c*t;

则lcm(a,b,c)=k/t;

若k/t=2^A;

则a,b,c中至少有一个数为2^A,至少有一个数是2^0,另外一个数为2^(0~A);共6*A种情况。

则，若k/t=2^A*3^B*5^C;

a,b,c的情况数为：(6*A)*(6*B)*(6*C);

GCD and LCM

Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65535/65535 K (Java/Others)

Total Submission(s): 1978 Accepted Submission(s): 876



Problem 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

Source

2013
ACM-ICPC吉林通化全国邀请赛——题目重现

#include <iostream>
#include <cstring>
using namespace std;
const int max_num = 1000000;

int vis[max_num],prim[max_num];
int len;
int sum;

void is_prim()
{
len = 0;
memset(vis,0,sizeof(vis));
memset(prim,0,sizeof(prim));
for(int i=2;i<max_num;i++)
{
if(!vis[i])
{
prim[len++] = i;
for(int j=2*i;j<max_num;j+=i)
{
vis[j] = 1;
}
}
}
}

int getPrim(int nn)
{
int i;
int s = 0;
sum = 1;
for(i=0;i<len && prim[i]*prim[i]<=nn;i++)
{
s=0;
while(nn%prim[i]==0)
{
s += 6;
nn/=prim[i];
}

if(s)
sum *= s;
}

if(nn > 1)
sum *= 6;
return sum;
}

int main()
{
int i,j,k,m,n;
int t;
int g,l;

is_prim();
cin>>t;
while(t--)
{
cin>>g>>l;
if(l%g)
{
cout<<"0"<<endl;
continue;
}

cout<<getPrim(l/g)<<endl;
}
return 0;
}