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 C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;
using static Input;

public class Prog
{
    private const int INF = 1000000007;
    private const long mod = 998244353;
    public static void Main()
    {
        long N = NextLong();
        int M = NextInt();
        Factorization F = new Factorization(M);
        var sieves = F.Run();
        long ans = 1;
        Dictionary<long, long> d = new Dictionary<long, long>();
        foreach (var i in sieves)
        {
            if (!d.ContainsKey(i)) d[i] = 0;
            d[i]++;
        }
        foreach (var e in d.Values)
        {
            //n H e
            long box = nHr(N, e) % INF;
            ans *= box;
            ans %= INF;
        }
        Console.WriteLine(ans);
    }
    private static long nHr(long n, long r)
    {
        long a = 1;
        for (long x = n + r - 1; x >= n; x--) { a *= x; a %= INF; }
        long b = 1;
        for (long x = 1; x <= r; x++) b *= x;
        b = RepPow(b, INF - 2);
        return (a * b);
    }
    public static long RepPow(long x, long p, long m = 1000000007)
    {
        if (p == 1) return x % m;
        long b = RepPow(x, p / 2, m) % m;
        if (p % 2 == 0) { return (b * b) % m; }
        else { return (x * ((b * b) % m)) % m; }
    }
}

public class Factorization
{
    private static long N;

    public Factorization(long n) { N = n; }

    //最大公約数を返す
    public static long GCD(long a, long b)
    {
        if (a < b) return GCD(b, a);
        long r;
        while (b != 0)
        {
            r = a % b;
            a = b;
            b = r;
        }
        return a;
    }

    //素数判定
    public static bool IsPrime(long n)
    {
        if (n < 2) return false;
        if (n == 2) return true;
        if (n % 2 == 0) return false;

        double root = Math.Sqrt(n);
        for (int i = 3; i <= root; i += 2)
        {
            if (n % i == 0) return false;
        }
        return true;
    }

    //試し割り
    private static int[] TryPrimes = new int[] { 2, 3, 5, 7, 11 };
    public long TryAllocation(long p)
    {
        long count = 0;
        while (N % p == 0)
        {
            count++;
            N /= p;
        }
        return count;
    }

    private int[] xi = new int[] { 13, 17, 19, 23, 29 };
    private long f(long x, int suffix)
    {
        return (xi[suffix % xi.Count()] * x + suffix) % N;
    }

    //nを分解
    public long PollardsRho(long n, int suffix = 0)
    {
        //素数を渡すと失敗するらしい
        if (IsPrime(n)) return n; //もう素数
        long x = 2, y = 2, d = 1;
        while (d == 1)
        {
            x = f(x, suffix);
            y = f(f(y, suffix), suffix);
            d = GCD(Math.Abs(x - y), n);
        }
        if (d == n)
        {
            //f(x)を変えて試す
            return PollardsRho(n, suffix + 1);
        }
        return d;
    }

    public IEnumerable<long> Run()
    {
        //小さな素因数の計算
        foreach (var prime in TryPrimes)
        {
            var box = TryAllocation(prime);
            for (int i = 0; i < box; i++) yield return prime;
        }

        //残りをRhoで解く
        while (true)
        {
            if (N == 1) break;
            var ret = PollardsRho(N);
            yield return ret;
            N /= ret;
        }
    }
}

public class Input
{
    private static Queue<string> queue = new Queue<string>();
    private static void Confirm() { if (queue.Count == 0) foreach (var s in Console.ReadLine().Split(' ')) queue.Enqueue(s); }
    public static int NextInt() { Confirm(); return int.Parse(queue.Dequeue()); }
    public static long NextLong() { Confirm(); return long.Parse(queue.Dequeue()); }
    public static string NextString() { Confirm(); return queue.Dequeue(); }
    public static double NextDouble() { Confirm(); return double.Parse(queue.Dequeue()); }
    public static int[] LineInt() { return Console.ReadLine().Split(' ').Select(int.Parse).ToArray(); }
    public static long[] LineLong() { return Console.ReadLine().Split(' ').Select(long.Parse).ToArray(); }
    public static string[] LineString() { return Console.ReadLine().Split(' ').ToArray(); }
    public static double[] LineDouble() { return Console.ReadLine().Split(' ').Select(double.Parse).ToArray(); }
}