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: Wrong Answer)

n,m = gets.split.map(&:to_i)
a = Hash.new(0)
$mod = 10**9+7
for i in 2..Math.sqrt(m).floor+1
  break if m < i*i
  while m%i == 0
    m /= i
    a[i] += 1
  end
end
if m != 1
  a[m] += 1
end
fact = Array.new(100100,1)
for i in 2..100099
  fact[i] = (fact[i-1]*i)%$mod
end
def exp(a,b)
  if b == 0
    1
  elsif b == 1
    a
  elsif b%2 == 0
    (exp(a,b/2)**2)%$mod
  else
    (a*exp(a,b/2)**2)%$mod
  end
end
fact_inv = Array.new(100100,0)
fact_inv[100099] = exp(fact[100099],$mod-2)
(100098).downto (1) do |i|
  fact_inv[i] = (fact_inv[i+1]*(i+1))%$mod
end
ans = 1
a.values.each do |i|
  ans *= (fact[n+i-1]*fact_inv[i]*fact_inv[n-1])%$mod
  ans %= $mod
end
puts ans