Problem:
Score : 400 points

Problem Statement

You are given positive integers N and M.

How many sequences a of length N consisting of positive integers satisfy a_1 \times a_2 \times ... \times a_N = M? Find the count modulo 10^9+7.

Here, two sequences a' and a'' are considered different when there exists some i such that a_i' \neq a_i''.

Constraints

All values in input are integers.

1 \leq N \leq 10^5

1 \leq M \leq 10^9

Input

Input is given from Standard Input in the following format:

N M

Output

Print the number of the sequences consisting of positive integers that satisfy the condition, modulo 10^9 + 7.

Sample Input 1

2 6

Sample Output 1

4

Four sequences satisfy the condition: \{a_1, a_2\} = \{1, 6\}, \{2, 3\}, \{3, 2\} and \{6, 1\}.

Sample Input 2

3 12

Sample Output 2

18

Sample Input 3

100000 1000000000

Sample Output 3

957870001

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <iostream>
#include <math.h>
#include <vector>
#include <map>
#include <stack>
#include <queue>
#include <set>
#include <algorithm>
#include <iomanip>
#include <string.h>

#define FOR(i,a,b) for(int i=(a);i<(b);++i)
#define REP(i,n)  FOR(i,0,n)
#define ALL(a)  (a).begin(),(a).end()

typedef long long lint;

using namespace std;

lint inv_mod(lint a,lint b,lint p){
  if(b==0)return 1;
  if(b%2==0){
    lint d=inv_mod(a,b/2,p);
    return (d*d)%p;
  }else{
    return (a*inv_mod(a,b-1,p))%p;
  }
}

int main(){
  int N,M;
  cin>>N>>M;
  int K=31624;
  // int K=10;
  lint p=1000*1000*1000+7;
  lint ans=1;
  int s[K]={0};
  FOR(i,2,K){
    while(M%i==0){
      if(M==0)break;
      M/=i;
      s[i]+=1;
    }
    if(M==0)break;
  }
  FOR(i,2,K){
    if(s[i]==0)continue;
    lint a=1;
    FOR(j,N,N+s[i]){
      a*=j;
      a%=p;
    }
    lint b=1;
    FOR(j,1,s[i]+1){
      b*=j;
      b%=p;
    }
    // cout<<s[i]<<a<<b<<endl;
    ans*=a;
    ans%=p;
    ans*=inv_mod(b,p-2,p);
    ans%=p;
  }
  cout<<ans<<endl;
  return 0;
}