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 Python (3.4.3) (judge verdict: Time Limit Exceeded)

from collections import defaultdict

MOD = 1000000007

mod_fact_dict = {}

def mod_fact(n):
    res = 1
    while n > 0:
        res *= n
        res %= MOD
        n -= 1
    return res

def duplicate_combination(n, k):
    t = n + k - 1
    if t not in mod_fact_dict:
        mod_fact_dict[t] = mod_fact(t)
    res = mod_fact_dict[t]
    res %= MOD
    if n not in mod_fact_dict:
        mod_fact_dict[n] = mod_fact(n)
    res *= mod_inv(mod_fact_dict[n])
    res %= MOD
    t = k - 1
    if t not in mod_fact_dict:
        mod_fact_dict[t] = mod_fact(t)
    res *= mod_inv(mod_fact_dict[t])
    res %= MOD
    return res

def _mod_inv(x, n):
    if n == 0:
        return 1
    t = _mod_inv(x, n >> 1)
    t = (t * t) % MOD
    return (t * (x if n & 1 else 1)) % MOD

def mod_inv(n):
    return _mod_inv(n, MOD - 2)

def primes(x):
    n = x
    for i in range(2, n + 1):
        if x == 1:
            return
        while x % i == 0:
            x //= i
            counter[i] += 1

if __name__ == '__main__':
    n, m = map(int, input().split())

    counter = defaultdict(int)
    primes(m)
    ans = 1
    d = {}
    for x in counter.values():
        if x not in d:
            d[x] = duplicate_combination(x, n)
        ans *= d[x]
        ans %= MOD
    print(ans)