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: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.Globalization;
using System.IO;
using System.Text;
using System.Linq;

using E = System.Linq.Enumerable;

internal partial class Solver {

    List<int> Factorize(int x) {
        var r = new List<int>();

        for (int i = 2; i * i <= x; i++) {
            int cnt = 0;
            while (x % i == 0) {
                x /= i;
                cnt++;
            }
            if (cnt > 0) r.Add(cnt);
        }

        if (x > 1) r.Add(1);

        return r;
    }

    public void Run() {
        var n = ni();
        var m = ni();
        int mod = 1000000007;

        var f = Factorize(m);

        var b = new Binomial(100000 * 2, mod);

        long ans = 1;

        foreach (var x in f) {
            ans *= b.H(x, n);
            ans %= mod;
        }

        cout.WriteLine(ans);
    }
}

public class Binomial {
    private readonly long[] factorial;
    private readonly long[] inverseFactorial;
    private readonly long[] inverse;
    private readonly int mod;

    public Binomial(int size, int primeMod) {
        size++;
        factorial = new long[size];
        inverseFactorial = new long[size];
        inverse = new long[size];
        mod = primeMod;
        Setup(size);
    }

    private void Setup(int size) {
        factorial[0] = factorial[1] = 1;
        inverseFactorial[0] = inverseFactorial[1] = 1;
        inverse[1] = 1;
        for (int i = 2; i < size; i++) {
            factorial[i] = factorial[i - 1] * i % mod;
            inverse[i] = (mod - (mod / i) * inverse[mod % i] % mod);
            inverseFactorial[i] = inverseFactorial[i - 1] * inverse[i] % mod;
        }
    }

    private long C(int s, int t) {
        if (s < 0 || t < 0 || s < t) {
            return 0;
        }

        if (t == 0 || s == t) {
            return 1;
        }

        if (s >= mod) {
            return C(s % mod, t % mod) * C(s / mod, t / mod) % mod; // Lucas' theorem
        }

        return factorial[s] * inverseFactorial[t] % mod * inverseFactorial[s - t] % mod;
    }

    public long this[int s, int t] {
        get {
            return C(s, t);
        }
    }

    public long P(int s, int t) {
        if (s < 0 || t < 0 || s < t) return 0;
        return factorial[s] * inverseFactorial[s - t] % mod;
    }

    public long H(int s, int t) {
        if (s < 0 || t < 0) return 0;
        if (s == 0 && t == 0) return 1;
        return C(s + t - 1, s);
    }

    public long H1(int s, int t) {
        return H(s - t, t);
    }
}
// PREWRITEN CODE BEGINS FROM HERE
internal partial class Solver : Scanner {
    public static void Main(string[] args) {
#if LOCAL
        byte[] inputBuffer = new byte[1000000];
        var inputStream = Console.OpenStandardInput(inputBuffer.Length);
        using (var reader = new StreamReader(inputStream, Console.InputEncoding, false, inputBuffer.Length)) {
            Console.SetIn(reader);
            new Solver(Console.In, Console.Out).Run();
        }
#else
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Solver(Console.In, Console.Out).Run();
        Console.Out.Flush();
#endif
    }

#pragma warning disable IDE0052
    private readonly TextReader cin;
    private readonly TextWriter cout;
#pragma warning restore IDE0052

    public Solver(TextReader reader, TextWriter writer)
        : base(reader) {
        cin = reader;
        cout = writer;
    }
    public Solver(string input, TextWriter writer)
        : this(new StringReader(input), writer) {
    }

#pragma warning disable IDE1006
#pragma warning disable IDE0051
    private int ni() { return NextInt(); }
    private int[] ni(int n) { return NextIntArray(n); }
    private long nl() { return NextLong(); }
    private long[] nl(int n) { return NextLongArray(n); }
    private double nd() { return NextDouble(); }
    private double[] nd(int n) { return NextDoubleArray(n); }
    private string ns() { return Next(); }
    private string[] ns(int n) { return NextArray(n); }
#pragma warning restore IDE1006
#pragma warning restore IDE0051
}

internal static class LinqPadExtension {
    [Conditional("DEBUG")]
    public static void Dump<T>(this T obj) {
#if DEBUG
        LINQPad.Extensions.Dump(obj);
#endif
    }
}

public class Scanner {
    private readonly TextReader Reader;
    private readonly Queue<string> TokenQueue = new Queue<string>();
    private readonly CultureInfo ci = CultureInfo.InvariantCulture;

    public Scanner()
        : this(Console.In) {
    }

    public Scanner(TextReader reader) {
        Reader = reader;
    }

    public int NextInt() { return int.Parse(Next(), ci); }
    public long NextLong() { return long.Parse(Next(), ci); }
    public double NextDouble() { return double.Parse(Next(), ci); }
    public string[] NextArray(int size) {
        string[] array = new string[size];
        for (int i = 0; i < size; i++) {
            array[i] = Next();
        }

        return array;
    }
    public int[] NextIntArray(int size) {
        int[] array = new int[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextInt();
        }

        return array;
    }

    public long[] NextLongArray(int size) {
        long[] array = new long[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextLong();
        }

        return array;
    }

    public double[] NextDoubleArray(int size) {
        double[] array = new double[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextDouble();
        }

        return array;
    }

    public string Next() {
        if (TokenQueue.Count == 0) {
            if (!StockTokens()) {
                throw new InvalidOperationException();
            }
        }
        return TokenQueue.Dequeue();
    }

    public bool HasNext() {
        if (TokenQueue.Count > 0) {
            return true;
        }

        return StockTokens();
    }

    private static readonly char[] _separator = new[] { ' ', '\t' };
    private bool StockTokens() {
        while (true) {
            string line = Reader.ReadLine();
            if (line == null) {
                return false;
            }

            string[] tokens = line.Split(_separator, StringSplitOptions.RemoveEmptyEntries);
            if (tokens.Length == 0) {
                continue;
            }

            foreach (string token in tokens) {
                TokenQueue.Enqueue(token);
            }

            return true;
        }
    }
}