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)

N,m = gets.split.map(&:to_i)
MOD = 1000000007
m0 = m
xs = Array.new(Math.sqrt(m)+2, 1)
xs[0] = xs[1] = 0
ds = {}
f = 2
while f <= Math.sqrt(m)
  if m % f == 0
    ds[f] = 0
    while m % f == 0
      ds[f] += 1
      m /= f
    end
  end
  f += 1
end
if m > 1
  ds[m] = 1
  m = 1
end
#p ds,m,ds.map{ |a,b| a**b }.inject(:*),m0

def d(a,b)
  dp = Array.new(b+1, 0)
  dp[0] = 1
  (0...a).each{
    ndp = Array.new(dp.size, 0)
    s = 0
    (0...dp.size).each{ |j|
      ndp[j] = (dp[j] + s) % MOD
      s = (s + dp[j]) % MOD
    }
    dp = ndp
#     p dp
  }
#  puts
  dp[-1]
end
ans = 1
ds.each{ |k,d| 
  v = d(N,d)
  ans = (ans * v) % MOD
}
p ans