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 Haskell (GHC 7.10.3) (judge verdict: Accepted)

{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE BinaryLiterals #-}
{-# LANGUAGE MultiWayIf #-}
{-# LANGUAGE NumDecimals #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TupleSections #-}

import Control.Applicative
import Control.Arrow
import Control.Monad
import Control.Monad.ST

import Data.Char
import Data.List
import Data.Maybe

import qualified Data.ByteString.Char8 as B

import Data.Array.ST.Safe
import Data.STRef

import Debug.Trace
import Text.Printf

readInt = readLn :: IO Int
readInts = map ( fst . fromJust . B.readInt ) . B.words <$> B.getLine
readNInts = readInt >>= flip replicateM readInt

which a b f = if f then a else b
mp [ a, b ] = ( a, b )

modBase = 1000000007 :: Integer

a *% b = ( a * b ) `mod` modBase

primeFactorization 1 = []
primeFactorization n = f 2 n
	where
		f p n
			| n == p = [p]
			| n `mod` p == 0 = p : f p ( n `div` p )
			| otherwise = f ( p + 1 ) n

nCr 0 _ = 1
nCr _ 0 = 1
nCr n r = nCr ( n - 1 ) ( r - 1 ) * n `div` r

nHr n r = nCr ( n + r - 1 ) r

main = do
	[ n, m ] <- map fromIntegral <$> readInts
	let degrees = map ( fromIntegral . length ) $ group $ primeFactorization m
	print $ foldl (*%) 1 $ map ( n `nHr` ) degrees