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.Text;
using System.Collections.Generic;
using System.Linq;
class Solve{
    public Solve(){}
    StringBuilder sb;
    ReadData re;
    public static int Main(){
        new Solve().Run();
        return 0;
    }
    void Run(){
        sb = new StringBuilder();
        re = new ReadData();
        Calc();
        Console.Write(sb.ToString());
    }
    void Calc(){
        int N = re.i();
        long M = re.l();
        PrimeFactor P = new PrimeFactor(100000);
        Fact F = new Fact(1000000);
        int[] D = P.PrimeDivide(M);
        int c = 1;
        long count = 1;
        for(int i=1;i<D.Length;i++){
            if(D[i] == D[i-1]){
                c++;
            }
            else{
                count *= F.GetConv(c+N-1,c);
                count %= Define.mod;
                c = 1;
            }
        }
        count *= F.GetConv(c+N-1,c);
        count %= Define.mod;
        sb.Append(count+"\n");
    }
}
class Fact{
    public long[] f;
    public long[] rf;
    public Fact(int N){
        f = new long[N+1];
        rf = new long[N+1];
        for(int i=0;i<N+1;i++){
            if(i == 0){
                f[i] = 1;
            }
            else{
                f[i] = (f[i-1]*i)%Define.mod;
            }
        }
        for(int i=N;i>=0;i--){
            if(i == N){
                rf[i] = pow(f[N],Define.mod-2);
            }
            else{
                rf[i] = rf[i+1]*(i+1)%Define.mod;
            }
        }
    }
    public long pow(long N,long K){
        if(K == 0){
            return 1;
        }
        else if(K % 2 == 0){
            long t = pow(N,K/2);
            return t*t%Define.mod;
        }
        else{
            return N*pow(N,K-1)%Define.mod;
        }
    }
    public long GetFact(int N){
        return f[N];
    }
    public long GetPerm(int N,int R){
        return f[N] * rf[N-R] % Define.mod;
    }
    public long GetConv(int N,int R){
        return ((f[N]*rf[R])%Define.mod*rf[N-R])%Define.mod;
    }
    public long GetRev(int N){
        if(N == 0){
            return 1;
        }
        else{
            return rf[N] * f[N-1] % Define.mod;
        }
    }
}
class PrimeFactor{
    int[] LowestPrimeFactor;
    int[] Prime;
    int N;
    public PrimeFactor(int N0){
        N = N0;
        N++;
        LowestPrimeFactor = new int[N];
        List<int> prime = new List<int>();
        for(int i=1;i<N;i++){
            LowestPrimeFactor[i] = i;
        }
        for(int i=2;i<N;i++){
            if(LowestPrimeFactor[i] == i){
                prime.Add(i);
            }
            for(int j=0;j<prime.Count;j++){
                if(prime[j]*i < N && prime[j] <= LowestPrimeFactor[i]){
                    LowestPrimeFactor[prime[j]*i] = prime[j];
                }
                else{
                    break;
                }
            }
        }
        Prime = prime.ToArray();
    }
    public int[] PrimeDivide(int X){
        List<int> ans = new List<int>();
        while(X != 1){
            ans.Add(LowestPrimeFactor[X]);
            X /= LowestPrimeFactor[X];
        }
        return ans.ToArray();
    }
    public int[] PrimeDivide(long X0){
        List<int> ans = new List<int>();
        int X = (int)X0;
        int i=0;
        while(X >= N){
            if(X % Prime[i] == 0){
                ans.Add(Prime[i]);
                X /= Prime[i];
            }
            else{
                i++;
                if(i == Prime.Length){
                    ans.Add(X);
                    X = 1;
                }
            }
        }
        while(X != 1){
            ans.Add(LowestPrimeFactor[X]);
            X /= LowestPrimeFactor[X];
        }
        return ans.ToArray();
    }
    public int[] Divisor(int X){
        List<int> ans = new List<int>();
        ans.Add(1);
        while(X != 1){
            int p = LowestPrimeFactor[X];
            int c = 0;
            while(X % p == 0){
                X /= p;
                c++;
            }
            int l = ans.Count;
            int d = p;
            for(int i=1;i<=c;i++){
                for(int j=0;j<l;j++){
                    ans.Add(d*ans[j]);
                }
                d *= p;
            }
        }
        return ans.ToArray();
    }
    public int[] Divisor(long X0){
        List<int> ans = new List<int>();
        int X = (int)X0;
        ans.Add(1);
        int i=0;
        while(X >= N){
            if(X % Prime[i] == 0){
                int p = Prime[i];
                int c = 0;
                while(X % p == 0){
                    X /= p;
                    c++;
                }
                int l = ans.Count;
                int d = p;
                for(int k=1;k<=c;k++){
                    for(int j=0;j<l;j++){
                        ans.Add(d*ans[j]);
                    }
                    d *= p;
                }
            }
            else{
                i++;
                if(i == Prime.Length){
                    int l = ans.Count;
                    for(int j=0;j<l;j++){
                        ans.Add(X*ans[j]);
                    }   
                    X = 1;
                }
            }
        }
        while(X != 1){
            int p = LowestPrimeFactor[X];
            int c = 0;
            while(X % p == 0){
                X /= p;
                c++;
            }
            int l = ans.Count;
            int d = p;
            for(int k=1;k<=c;k++){
                for(int j=0;j<l;j++){
                    ans.Add(d*ans[j]);
                }
                d *= p;
            }
        }
        return ans.ToArray();
    }
    public int Phi(int X){
        int ans = X;
        while(X != 1){
            int p = LowestPrimeFactor[X];
            while(X % p == 0){
                X /= p;
            }
            ans /= p;
            ans *= p-1;
        }
        return ans;
    }
    public int Phi(long X0){
        int ans = (int)X0;
        int X = (int)X0;
        int i=0;
        while(X >= N){
            if(X % Prime[i] == 0){
                int p = Prime[i];
                while(X % p == 0){
                    X /= p;
                }
                ans /= p;
                ans *= p-1;
            }
            else{
                i++;
                if(i == Prime.Length){
                    int p = X;
                    ans /= p;
                    ans *= p-1;
                    X = 1;
                }
            }
        }
        while(X != 1){
            int p = LowestPrimeFactor[X];
            while(X % p == 0){
                X /= p;
            }
            ans /= p;
            ans *= p-1;
        }
        return ans;
    }
}
class ReadData{
    string[] str;
    int counter;
    public ReadData(){
        counter = 0;
    }
    public string s(){
        if(counter == 0){
            str = Console.ReadLine().Split(' ');
            counter = str.Length;
        }
        counter--;
        return str[str.Length-counter-1];
    }
    public int i(){
        return int.Parse(s());
    }
    public long l(){
        return long.Parse(s());
    }
    public double d(){
        return double.Parse(s());
    }
    public int[] ia(int N){
        int[] ans = new int[N];
        for(int j=0;j<N;j++){
            ans[j] = i();
        }
        return ans;
    }
    public int[] ia(){
        str = Console.ReadLine().Split(' ');
        counter = 0;
        int[] ans = new int[str.Length];
        for(int j=0;j<str.Length;j++){
            ans[j] = int.Parse(str[j]);
        }
        return ans;
    }
    public long[] la(int N){
        long[] ans = new long[N];
        for(int j=0;j<N;j++){
            ans[j] = l();
        }
        return ans;
    }
    public long[] la(){
        str = Console.ReadLine().Split(' ');
        counter = 0;
        long[] ans = new long[str.Length];
        for(int j=0;j<str.Length;j++){
            ans[j] = long.Parse(str[j]);
        }
        return ans;
    }
    public double[] da(int N){
        double[] ans = new double[N];
        for(int j=0;j<N;j++){
            ans[j] = d();
        }
        return ans;
    }
    public double[] da(){
        str = Console.ReadLine().Split(' ');
        counter = 0;
        double[] ans = new double[str.Length];
        for(int j=0;j<str.Length;j++){
            ans[j] = double.Parse(str[j]);
        }
        return ans;
    }
    public List<int>[] g(int N,int[] f,int[] t){
        List<int>[] ans = new List<int>[N];
        for(int j=0;j<N;j++){
            ans[j] = new List<int>();
        }
        for(int j=0;j<f.Length;j++){
            ans[f[j]].Add(t[j]);
            ans[t[j]].Add(f[j]);
        }
        return ans;
    }
    public List<int>[] g(int N,int M){
        List<int>[] ans = new List<int>[N];
        for(int j=0;j<N;j++){
            ans[j] = new List<int>();
        }
        for(int j=0;j<M;j++){
            int f = i()-1;
            int t = i()-1;
            ans[f].Add(t);
            ans[t].Add(f);
        }
        return ans;
    }
    public List<int>[] g(){
        int N = i();
        int M = i();
        List<int>[] ans = new List<int>[N];
        for(int j=0;j<N;j++){
            ans[j] = new List<int>();
        }
        for(int j=0;j<M;j++){
            int f = i()-1;
            int t = i()-1;
            ans[f].Add(t);
            ans[t].Add(f);
        }
        return ans;
    }
}
public static class Define{
    public const long mod = 1000000007;
}
public static class Debug{
    public static void Print(double[,,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                for(int l=0;l<k.GetLength(2);l++){
                    Console.Write(k[i,j,l]+" ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
        }
    }
    public static void Print(double[,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                Console.Write(k[i,j]+" ");
            }
            Console.WriteLine();
        }
    }
    public static void Print(double[] k){
        for(int i=0;i<k.Length;i++){
            Console.WriteLine(k[i]);
        }
    }
    public static void Print(long[,,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                for(int l=0;l<k.GetLength(2);l++){
                    Console.Write(k[i,j,l]+" ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
        }
    }
    public static void Print(long[,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                Console.Write(k[i,j]+" ");
            }
            Console.WriteLine();
        }
    }
    public static void Print(long[] k){
        for(int i=0;i<k.Length;i++){
            Console.WriteLine(k[i]);
        }
    }
    public static void Print(int[,,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                for(int l=0;l<k.GetLength(2);l++){
                    Console.Write(k[i,j,l]+" ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
        }
    }
    public static void Print(int[,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                Console.Write(k[i,j]+" ");
            }
            Console.WriteLine();
        }
    }
    public static void Print(int[] k){
        for(int i=0;i<k.Length;i++){
            Console.WriteLine(k[i]);
        }
    }
}