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)

import math
def factorize(m):
    fct = []
    b = 2
    while b * b <= m:
        cnt = 0
        while m % b == 0:
            m //= b
            cnt += 1
        if b > 0:
            fct.append([b, cnt])
        b += 1 
    if m > 1:
        fct.append([m, 1])
    return fct

def comb(n, r):
    if n < r or n < 0 or r < 0:
        return 0
    else:
        return math.factorial(n)//(math.factorial(n-r)*math.factorial(r))

N, M = map(int, input().split())
ans = 1
M_fct = factorize(M)

for mf in M_fct:
    #mf[1]個をN通りに分ける→comb(mf[1]+N-1,mf[1])
    ans *= comb(N + mf[1] - 1, mf[1])
print(ans% (10 ** 9 + 7))