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)
MOD = 1000000007
m0 = m
xs = Array.new(Math.sqrt(m)+2, 1)
xs[0] = xs[1] = 0
ps = []
(2...xs.size).each{ |f|
  next if xs[f] == 0
  ps << f
  k = f*2
  while k < xs.size
    xs[k] = 0
    k += f
  end
}

ds = {}
ps.each{ |k|
  next if m % k > 0
  ds[k] = 0
  while m % k == 0
    ds[k] += 1
    m /= k
  end
}
#p ds,m
if m > 1
  # m == m0
  p N
else
  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 += dp[j]
      }
      dp = ndp
 #     p dp
    }
  #  puts
    dp[-1]
  end
  ans = 1
  ds.each{ |k,d| 
  #  p [k,d]
    v = d(N,d)
    ans = (ans * v) % MOD
  }
  p ans
end