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: Accepted)

#include <iostream>
using namespace std;

uint64_t combination(uint64_t, uint64_t, const uint64_t&);

int main()
{
  const uint64_t mod = 1'000'000'007;
  uint64_t n, m;
  cin >> n >> m;
  uint64_t mm = m;
  uint64_t ans = 1;
  for (auto i = 2; i * i <= mm; i++) {
    if (mm % i == 0) {
      uint64_t count = 0;
      while (mm % i == 0) {
	count++;
	mm /= i;
      }
      ans *= combination(n + count - 1, count, mod);
      ans %= mod;
    }
  }
  if (mm != 1) {
    ans *= n;
    ans %= mod;
  }
  cout << ans << endl;
}

uint64_t modpow(uint64_t a, uint64_t p, const uint64_t& mod)
{
  if (p == 0) return 1;
  if (p % 2 == 0) {
    uint64_t h = modpow(a, p / 2, mod);
    return h * h % mod;
  } else {
    return a * modpow(a, p - 1, mod) % mod;
  }
}

uint64_t combination(uint64_t a, uint64_t b, const uint64_t& mod)
{
  if (a - b < b) return combination(a, a - b, mod);
  uint64_t ansmul = 1;
  uint64_t ansdiv = 1;
  for (uint64_t i = 0; i < b; i++) {
    ansmul *= a - i;
    ansdiv *= i + 1;
    ansmul %= mod;
    ansdiv %= mod;
  }
  uint64_t ans = ansmul * modpow(ansdiv, mod - 2, mod) % mod;
  return ans;
}