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 Rust (1.42.0) (judge verdict: Accepted)

use std::io;
use std::str::FromStr;
use std::collections::HashMap;

static MOD: u64 = 1_000_000_007;

fn input_tuple() -> (u64, u64) {
    let mut line: String = String::new();
    io::stdin().read_line(&mut line);
    let vec: Vec<u64> = line.split_whitespace().map(|s| s.parse().unwrap()).collect();
    (vec[0], vec[1])
}

fn prime_factors(n: u64) -> HashMap<u64, u64> {
    let limit = n;
    let mut n = n; 
    let mut prime_numbers: HashMap<u64, u64> = HashMap::new();
    let mut candidate = 2;
    while n > 1 {
        if candidate * candidate > limit {
            prime_numbers.insert(n, 1);
            return prime_numbers;
        }
        while n % candidate == 0 {
            let val = prime_numbers.entry(candidate).or_insert(0);
            *val += 1;
            n /= candidate;
        }
        candidate += 1;
    }
    return prime_numbers;
}

struct CombinationCalculator {
    f: Vec<u64>,
    finv: Vec<u64>,
}

impl CombinationCalculator {
    fn new(n: usize) -> CombinationCalculator {
        let mut f = vec![1u64; n];
        let mut inv = vec![1u64; n];
        let mut finv = vec![1u64; n];
        for i in 2..n {
            f[i] = f[i - 1] * i as u64 % MOD;
            inv[i] = MOD - inv[MOD as usize % i] * (MOD / i as u64) % MOD;
            finv[i] = finv[i - 1] * inv[i] % MOD;
        }
        CombinationCalculator {f: f, finv: finv}
    }

    fn calc(&self, n: u64, m: u64) -> u64 {
        if n < m {
            return 0;
        } 
        self.f[n as usize] * (self.finv[m as usize] * self.finv[(n - m) as usize] % MOD) % MOD
    }
}

fn main() {
    let (n, m) = input_tuple();
    let factor_map = prime_factors(m);
    let combination_calculator: CombinationCalculator = CombinationCalculator::new(n as usize + 30);
    let mut sum = 1;
    for v in factor_map.values() {
        sum *= combination_calculator.calc(v + n - 1, *v);
        sum %= MOD;
    }
    println!("{}", sum);
}