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.IO;
using System.Linq;
using System.Numerics;
using System.Text;
using System.Threading.Tasks;

namespace AtCorder
{
	public class Program
	{
		static void Main(string[] args)
		{
			var cin = new Scanner2();
			int n = cin.Int();
			int m = cin.Int();
			
			var factors = GetPrimeFactors(m);
			if (factors.Count == 0) {
				Console.WriteLine(1);
				return;
			}

			long max = factors.Max(x => x.Value);
			ModCounting.InitializeFactorial(n + max);

			ModInt ans = 1;
			foreach (long key in factors.Keys) {
				ans *= ModCounting.HomogeneousProduct(n, factors[key]);
			}

			Console.WriteLine(ans.Value);
		}

		public static Dictionary<long, long> GetPrimeFactors(long value)
		{
			var factors = new Dictionary<long, long>();
			for (long i = 2; i * i <= value; ++i) {
				while (value % i == 0) {
					if (factors.ContainsKey(i) == false) {
						factors[i] = 1;
					} else {
						factors[i] += 1;
					}

					value /= i;
				}
			}

			if (value != 1) {
				factors[value] = 1;
			}

			return factors;
		}
	}

	public struct ModInt
	{
		public static long Mod { get; set; } = (long)1e9 + 7;

		public static ModInt Pow(ModInt value, long k)
		{
			return Pow(value.Value, k);
		}

		public static ModInt Pow(long value, long k)
		{
			long ret = 1;
			for (k %= Mod - 1; k > 0; k >>= 1, value = value * value % Mod) {
				if ((k & 1) == 1) {
					ret = ret * value % Mod;
				}
			}
			return new ModInt(ret);
		}

		public static ModInt Inverse(ModInt value)
		{
			return Pow(value, Mod - 2);
		}

		public long Value;

		public ModInt(long value)
		{
			value %= Mod;
			if (value < 0) {
				value += Mod;
			}

			Value = value;
		}

		public static ModInt operator +(ModInt lhs, ModInt rhs)
		{
			lhs.Value += rhs.Value;
			if (lhs.Value >= Mod) {
				lhs.Value -= Mod;
			}
			return lhs;
		}

		public static ModInt operator -(ModInt lhs, ModInt rhs)
		{
			lhs.Value -= rhs.Value;
			if (lhs.Value < 0) {
				lhs.Value += Mod;
			}

			return lhs;
		}

		public static ModInt operator *(ModInt lhs, ModInt rhs)
		{
			return new ModInt(lhs.Value * rhs.Value % Mod);
		}
		public static ModInt operator /(ModInt lhs, ModInt rhs)
		{
			long exp = Mod - 2;
			while (exp > 0) {
				if (exp % 2 > 0) {
					lhs *= rhs;
				}

				rhs *= rhs;
				exp /= 2;
			}

			return lhs;
		}

		public static implicit operator ModInt(long n)
		{
			n %= Mod;
			if (n < 0) {
				n += Mod;
			}

			return new ModInt(n);
		}

		public override string ToString()
		{
			return Value.ToString();
		}
	}

	public static class ModCounting
	{
		private static long p_;

		private static long[] factorial_;
		private static long[] inverseFactorial_;
		private static long[] inverse_;

		public static void InitializeFactorial(long max, long p = 1000000007)
		{
			p_ = p;

			factorial_ = new long[max + 1];
			inverseFactorial_ = new long[max + 1];
			inverse_ = new long[max + 1];

			factorial_[0] = factorial_[1] = 1;
			inverseFactorial_[0] = inverseFactorial_[1] = 1;
			inverse_[1] = 1;
			for (int i = 2; i <= max; i++) {
				factorial_[i] = factorial_[i - 1] * i % p;
				inverse_[i] = p - inverse_[p % i] * (p / i) % p;
				inverseFactorial_[i] = inverseFactorial_[i - 1] * inverse_[i] % p;
			}
		}

		public static long Factorial(long n)
		{
			if (n < 0) {
				return 0;
			}

			return factorial_[n];
		}

		public static long InverseFactorial(long n)
		{
			if (n < 0) {
				return 0;
			}

			return inverseFactorial_[n];
		}

		public static long Inverse(long n)
		{
			if (n < 0) {
				return 0;
			}

			return inverse_[n];
		}

		public static long Permutation(long n, long k)
		{
			if (n < k || (n < 0 || k < 0)) {
				return 0;
			}

			return factorial_[n] * inverseFactorial_[n - k] % p_;
		}

		public static long RepeatedPermutation(long n, long k)
		{
			long ret = 1;
			for (k %= p_ - 1; k > 0; k >>= 1, n = n * n % p_) {
				if ((k & 1) == 1) {
					ret = ret * n % p_;
				}
			}

			return ret;
		}

		public static long Combination(long n, long k)
		{
			if (n < k || (n < 0 || k < 0)) {
				return 0;
			}

			return factorial_[n] * (inverseFactorial_[k] * inverseFactorial_[n - k] % p_) % p_;
		}

		public static long HomogeneousProduct(long n, long k)
		{
			if (n < 0 || k < 0) {
				return 0;
			}

			return Combination(n + k - 1, k);
		}
	}


	public class Scanner2
	{
		private readonly char[] delimiter_ = new char[] { ' ' };
		private readonly string filePath_;
		private readonly Func<string> reader_;
		private string[] buf_;
		private int index_;

		public Scanner2(string file = "")
		{
			if (string.IsNullOrWhiteSpace(file)) {
				reader_ = Console.ReadLine;
			} else {
				filePath_ = file;
				var fs = new StreamReader(file);
				reader_ = fs.ReadLine;
			}
			buf_ = new string[0];
			index_ = 0;
		}

		public string Next()
		{
			if (index_ < buf_.Length) {
				return buf_[index_++];
			}

			string st = reader_();
			while (st == "") {
				st = reader_();
			}

			buf_ = st.Split(delimiter_, StringSplitOptions.RemoveEmptyEntries);
			if (buf_.Length == 0) {
				return Next();
			}

			index_ = 0;
			return buf_[index_++];
		}

		public int Int() => int.Parse(Next());
		public long Long() => long.Parse(Next());
		public double Double() => double.Parse(Next());

		public int[] ArrayInt(int N, int add = 0)
		{
			int[] Array = new int[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Int() + add;
			}
			return Array;
		}

		public long[] ArrayLong(int N, long add = 0)
		{
			long[] Array = new long[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Long() + add;
			}
			return Array;
		}

		public double[] ArrayDouble(int N, double add = 0)
		{
			double[] Array = new double[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Double() + add;
			}
			return Array;
		}

		public void Save(string text)
		{
			if (string.IsNullOrWhiteSpace(filePath_)) {
				return;
			}

			File.WriteAllText(filePath_ + "_output.txt", text);
		}
	}
}