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: Runtime Error)

#require 'pp'
require 'prime'

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

primes = Prime.prime_division(m)

class Fractional
    attr_reader :mod

    def initialize(max, mod)
        @max = max
        @mod = mod
        @frac = {
            0 => 1,
            1 => 1,
        }
        @frac_inv = {
            0 => 1,
            1 => 1,  
        }
        @inv = {
            1 => 1,
        }
        pre_calc(max)
    end

    def pre_calc(max)
        2.upto(max) do |i|
            @frac[i] = @frac[i - 1] * i % @mod;
            @inv[i] = @mod - @inv[@mod % i] * (@mod / i) % @mod
            @frac_inv[i] = @frac_inv[i - 1] * @inv[i] % @mod
        end
    end

    def at(i)
        @frac[i]
    end

    def inv_at(i)
        @frac_inv[i]
    end
end

class Combination
    def initialize(fraction)
        @fraction = fraction
    end

    def calc(n, r)
        return 0 if n < r
        return 0 if (n < 0 || r < 0)

        denom_inv = (@fraction.inv_at(r) * @fraction.inv_at(n - r) % @fraction.mod)
        return @fraction.at(n) * denom_inv % @fraction.mod
    end
end

f = Fractional.new(10 ** 5 + 10, 10 ** 9 + 7)
c = Combination.new(f)

total = 1
primes.each do |prime, num|
    # 同一素数の数
    x = num
    # 仕切りの数
    y = (n-1)

    total = total * c.calc(x + y, x) % f.mod
end

puts total