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 Ruby (2.3.3) (judge verdict: Accepted)

require 'prime'

n, m = gets.strip.split.map(&:to_i)
MOD = 10**9+7

def power(n, k)
  return 1 if k==0
  return power(n, k/2)**2 % MOD if k.even?
  n*power(n, k-1) % MOD
end

fmax = Math.sqrt(m).ceil+n
@factor = [1,1]
(2..fmax).each do |i|
  @factor[i] = i*@factor[i-1] % MOD
end

@ifactor = [1,1]
@ifactor[fmax-2] = power(@factor[fmax-2], MOD-2) % MOD
(fmax-3).downto(2).each do |i|
  @ifactor[i] = @ifactor[i+1]*(i+1) % MOD
end

def comb(n, k)
  @factor[n]*(@ifactor[n-k]*@ifactor[k]%MOD)%MOD
end

divisions = m.prime_division

ans=1
divisions.each do |pr, div|
  ans = ans * comb(div+n-1, n-1) % MOD
end

#p @factor
#p @ifactor

p ans