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.Text;
using System.Collections.Generic;
using System.Linq;
using System.Collections;

namespace Equeue
{
    
    
    class Equeue
    {
        static Dictionary<long,long> Primes;
        
        static void Main(string[] args)
        {
            var NM = Console.ReadLine().Split(' ').Select(x => long.Parse(x)).ToArray();
            var N = NM[0];
            var M = NM[1];
          
            Primes = new Dictionary<long,long>();
            
            Factorization(M);
            
            long ans = 1;
            //N個のものからR個選ぶ組み合わせ
            var kaijo = new long[1000000];
            //階乗のモジュラー逆数
            var Ikaijo = new long[1000000];
            kaijo[0] = 1;
            Ikaijo[0] = 1;
            
            for(long i = 1;i < 1000000 ;i++){
                kaijo[i] = (kaijo[i-1]*i)%1000000007;
                Ikaijo[i] = RepeatSquaring(kaijo[i]);
            }
            long mo = 1000000007;
            
            
            
            foreach(KeyValuePair<long, long> item in Primes) {
                
                //ans += kaijo[N+item.Value-1]/(kaijo[N-1]*kaijo[item.Value]);
                
                ans *= (((kaijo[N+item.Value-1]%mo)*(Ikaijo[item.Value]%mo)%mo)*Ikaijo[N-1])%mo;
                ans %= mo;
            }

            Console.WriteLine(ans);
        }  
        
        
        
        //繰り返し2乗法
        static long RepeatSquaring(long N,long M = 1000000005,long mod = 1000000007){
            if(M == 0){
                return 1;
            }else if(M%2 == 0){
                long t = RepeatSquaring(N,(M/2),mod)%mod;
                return (t*t)%mod;
            }else{
                return ((N%mod)*RepeatSquaring(N,M-1,mod))%mod;
            }
            
            }
            
            
            
        static void Factorization(long n){
            long i = 2;
            long tmp = n;
            
            while(i*i <= n){
                if(tmp%i == 0){
                    if(Primes.ContainsKey(i)){
                        Primes[i]++;
                    }else{
                        Primes.Add(i,1);
                    }
                    tmp /= i;
                }else{
                    i++;
                }
            }
            
            if(tmp != 1){
                Primes.Add(tmp,1);
            }
            
            
        }
        
      
      
      
    }
  


    public class Node{
      public long loc = long.MinValue;
      public Dictionary<long,long> adj = new Dictionary<long,long>();
    }
  
  
    public class PriorityQueue<T> : IEnumerable<T>
    {
        private readonly List<T> _data = new List<T>();
        private readonly IComparer<T> _comparer;
        private readonly bool _isDescending;

        public PriorityQueue(IComparer<T> comparer, bool isDescending = true)
        {
            _comparer = comparer;
            _isDescending = isDescending;
        }

        public PriorityQueue(Comparison<T> comparison, bool isDescending = true)
            : this(Comparer<T>.Create(comparison), isDescending)
        {
        }

        public PriorityQueue(bool isDescending = true)
            : this(Comparer<T>.Default, isDescending)
        {
        }

        public void Enqueue(T item)
        {
            _data.Add(item);
            var childIndex = _data.Count - 1;
            while (childIndex > 0)
            {
                var parentIndex = (childIndex - 1) / 2;
                if (Compare(_data[childIndex], _data[parentIndex]) >= 0)
                    break;
                Swap(childIndex, parentIndex);
                childIndex = parentIndex;
            }
        }

        public T Dequeue()
        {
            var lastIndex = _data.Count - 1;
            var firstItem = _data[0];
            _data[0] = _data[lastIndex];
            _data.RemoveAt(lastIndex--);
            var parentIndex = 0;
            while (true)
            {
                var childIndex = parentIndex * 2 + 1;
                if (childIndex > lastIndex)
                    break;
                var rightChild = childIndex + 1;
                if (rightChild <= lastIndex && Compare(_data[rightChild], _data[childIndex]) < 0)
                    childIndex = rightChild;
                if (Compare(_data[parentIndex], _data[childIndex]) <= 0)
                    break;
                Swap(parentIndex, childIndex);
                parentIndex = childIndex;
            }
            return firstItem;
        }

        public T Peek()
        {
            return _data[0];
        }

        private void Swap(int a, int b)
        {
            var tmp = _data[a];
            _data[a] = _data[b];
            _data[b] = tmp;
        }

        private int Compare(T a, T b)
        {
            return _isDescending ? _comparer.Compare(b, a) : _comparer.Compare(a, b);
        }

        public int Count => _data.Count;

        public IEnumerator<T> GetEnumerator()
        {
            return _data.GetEnumerator();
        }

        IEnumerator IEnumerable.GetEnumerator() => GetEnumerator();
    }
}