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 (1.15.1) (judge verdict: Runtime Error)

#![allow(non_snake_case, unused)]
use std::cmp::*;
use std::collections::*;
use std::f64::consts::*;

macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        let mut next = || { iter.next().unwrap() };
        input_inner!{next, $($r)*}
    };
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};

    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };

    ($next:expr, mut $var:ident : $t:tt $($r:tt)*) => {
        let mut $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, [ $t:tt ]) => {
        {
            let len = read_value!($next, usize);
            (0..len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
        }
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, bytes) => {
        read_value!($next, String).into_bytes()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

fn prime_factor(n: usize) -> HashMap<usize, usize> {
    let mut res = HashMap::new();
    let mut n_ = n.clone();

    for i in 2..n + 1 {
        if i * i > n_ {
            break;
        }
        while (n_ % i == 0) {
            *res.entry(i).or_insert(0) += 1;
            n_ /= i;
        }
    }
    if (n_ != 1) {
        res.insert(n_, 1);
    }
    res
}

struct Comb {
    fac: Vec<u64>,
    finv: Vec<u64>,
    inv: Vec<u64>,
    m: u64,
}

impl Comb {
    fn new(n: usize, m: u64) -> Comb {
        let mut fac = vec![0u64; n + 1];
        let mut finv = vec![0u64; n + 1];
        let mut inv = vec![0u64; n + 1];

        fac[0] = 1;
        fac[1] = 1;
        finv[0] = 1;
        finv[1] = 1;
        inv[1] = 1;

        for i in 2..n + 1 {
            fac[i] = fac[(i - 1) as usize] * i as u64 % m;
            inv[i] = m - inv[(m % i as u64) as usize] * (m / i as u64) % m;
            finv[i] = finv[(i - 1) as usize] * inv[i] % m;
        }
        Comb {
            fac: fac,
            finv: finv,
            inv: inv,
            m: m,
        }
    }
    fn calc(&self, n: usize, k: usize) -> u64 {
        if n < k {
            return 0;
        }

        return self.fac[n] * (self.finv[k] * self.finv[n - k] % self.m) % self.m;
    }
}

const MOD: u64 = 1_000_000_000 + 7;
fn main() {
    input! {
        n: usize, m: usize,
    }
    let pf = prime_factor(m);

    let comb = Comb::new(n + 10, MOD);;
    let mut ans = 1;
    for (&k, &v) in &pf {
        ans *= comb.calc(v + n - 1, v);
        ans %= MOD;
    }

    println!("{}", ans);
}