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)

MOD = 10 ** 9 + 7

def eratosthnes(n)
  n = (n ** 0.5).to_i
  prime = []
  search = Array.new(n+1,true)
  2.upto(n) do |i|
    if search[i]
      prime << i
      m = i ** 2
      until m > n
        search[m] = false
        m += i
      end
    end
  end
  return prime
end

def fact(n)
  @cache ||= []
  @cache[n] ||= (n==0) ? 1 : (n * fact(n-1)) % MOD;
end

def pow(x,n)
  ret = 1
  while n > 0
    ret = (ret * x) % MOD if n&1 == 1
    n >>= 1
    x = (x*x) % MOD
  end
  return ret
end

def combin(n,k)
  ret = (fact(n) * pow(fact(n-k),MOD-2)) % MOD
  return (ret * pow(fact(k),MOD-2)) % MOD
end

n,m = gets.split.map(&:to_i)

n.times do |i|
  fact(i)
end
prime = eratosthnes(m)

exp = []
prime.each do |k|
  e = 0
  while m % k == 0
    m /= k
    e += 1
  end
  exp << e if e != 0
end
exp << 1 if m != 1

ans = 1
exp.each do |k|
  ans = (ans * combin(n+k-1,k)) % MOD
end
p ans