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: Wrong Answer)

#[doc = " https://github.com/hatoo/competitive-rust-snippets"]
#[allow(unused_imports)]
use std::cmp::{max, min, Ordering};
#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
#[allow(unused_imports)]
use std::io::{stdin, stdout, BufWriter, Write};
#[allow(unused_imports)]
use std::iter::FromIterator;
#[allow(unused_macros)]
macro_rules ! debug { ( $ ( $ a : expr ) ,* ) => { eprintln ! ( concat ! ( $ ( stringify ! ( $ a ) , " = {:?}, " ) ,* ) , $ ( $ a ) ,* ) ; } }
#[macro_export]
macro_rules ! input { ( source = $ s : expr , $ ( $ r : tt ) * ) => { let mut parser = Parser :: from_str ( $ s ) ; input_inner ! { parser , $ ( $ r ) * } } ; ( parser = $ parser : ident , $ ( $ r : tt ) * ) => { input_inner ! { $ parser , $ ( $ r ) * } } ; ( new_stdin_parser = $ parser : ident , $ ( $ r : tt ) * ) => { let stdin = std :: io :: stdin ( ) ; let reader = std :: io :: BufReader :: new ( stdin . lock ( ) ) ; let mut $ parser = Parser :: new ( reader ) ; input_inner ! { $ parser , $ ( $ r ) * } } ; ( $ ( $ r : tt ) * ) => { input ! { new_stdin_parser = parser , $ ( $ r ) * } } ; }
#[macro_export]
macro_rules ! input_inner { ( $ parser : ident ) => { } ; ( $ parser : ident , ) => { } ; ( $ parser : ident , $ var : ident : $ t : tt $ ( $ r : tt ) * ) => { let $ var = read_value ! ( $ parser , $ t ) ; input_inner ! { $ parser $ ( $ r ) * } } ; }
#[macro_export]
macro_rules ! read_value { ( $ parser : ident , ( $ ( $ t : tt ) ,* ) ) => { ( $ ( read_value ! ( $ parser , $ t ) ) ,* ) } ; ( $ parser : ident , [ $ t : tt ; $ len : expr ] ) => { ( 0 ..$ len ) . map ( | _ | read_value ! ( $ parser , $ t ) ) . collect ::< Vec < _ >> ( ) } ; ( $ parser : ident , chars ) => { read_value ! ( $ parser , String ) . chars ( ) . collect ::< Vec < char >> ( ) } ; ( $ parser : ident , usize1 ) => { read_value ! ( $ parser , usize ) - 1 } ; ( $ parser : ident , $ t : ty ) => { $ parser . next ::<$ t > ( ) . expect ( "Parse error" ) } ; }
use std::io;
use std::io::BufRead;
use std::str;
pub struct Parser<R> {
    reader: R,
    buf: Vec<u8>,
    pos: usize,
}
impl Parser<io::Empty> {
    pub fn from_str(s: &str) -> Parser<io::Empty> {
        Parser {
            reader: io::empty(),
            buf: s.as_bytes().to_vec(),
            pos: 0,
        }
    }
}
impl<R: BufRead> Parser<R> {
    pub fn new(reader: R) -> Parser<R> {
        Parser {
            reader: reader,
            buf: vec![],
            pos: 0,
        }
    }
    pub fn update_buf(&mut self) {
        self.buf.clear();
        self.pos = 0;
        loop {
            let (len, complete) = {
                let buf2 = self.reader.fill_buf().unwrap();
                self.buf.extend_from_slice(buf2);
                let len = buf2.len();
                if len == 0 {
                    break;
                }
                (len, buf2[len - 1] <= 0x20)
            };
            self.reader.consume(len);
            if complete {
                break;
            }
        }
    }
    pub fn next<T: str::FromStr>(&mut self) -> Result<T, T::Err> {
        loop {
            let mut begin = self.pos;
            while begin < self.buf.len() && (self.buf[begin] <= 0x20) {
                begin += 1;
            }
            let mut end = begin;
            while end < self.buf.len() && (self.buf[end] > 0x20) {
                end += 1;
            }
            if begin != self.buf.len() {
                self.pos = end;
                return str::from_utf8(&self.buf[begin..end]).unwrap().parse::<T>();
            } else {
                self.update_buf();
            }
        }
    }
}
#[allow(unused_macros)]
macro_rules ! debug { ( $ ( $ a : expr ) ,* ) => { eprintln ! ( concat ! ( $ ( stringify ! ( $ a ) , " = {:?}, " ) ,* ) , $ ( $ a ) ,* ) ; } }
#[doc = " https://github.com/hatoo/competitive-rust-snippets"]
const BIG_STACK_SIZE: bool = true;
#[allow(dead_code)]
fn main() {
    use std::thread;
    if BIG_STACK_SIZE {
        thread::Builder::new()
            .stack_size(32 * 1024 * 1024)
            .name("solve".into())
            .spawn(solve)
            .unwrap()
            .join()
            .unwrap();
    } else {
        solve();
    }
}
fn factorial(a: usize) -> usize {
    if a == 0 {
        return 1
    }
    let mut n = 1;
    let mut a = a;
    while a > 1 {
        n *= a;
        a -= 1;
    }
    n
}
fn nPk(a: usize, b: usize) -> usize {
    assert!(a>=b);
    factorial(a) / factorial(a-b)
}
fn nCk(a: usize, b: usize) -> usize {
    factorial(a) / factorial(a-b) / factorial(b)
}
fn nHk(a: usize, b: usize) -> usize {
    nCk(a+b-1, b)
}
// O(root(n))
fn root_int(n: usize) -> usize {
    let mut d = 1;
    while d * d <= n {
        d += 1;
    }
    d - 1
}
fn prime_factors(n: usize) -> std::collections::HashMap<usize,usize> {
    let mut n = n;
    let mut m = std::collections::HashMap::new();
    for i in 2..root_int(n)+1 {
        while n % i == 0 {
            if !m.contains_key(&i) {
                m.insert(i, 0 as usize);
            }
            *m.get_mut(&i).unwrap() += 1;
            n /= i;
        }
    }
    if n != 1 {
        m.insert(n, 1);
    }
    m
}
fn solve() {
    input! {
        N: usize, M: usize,
    }
    let p: usize = 1_000_000_007;
    let mut m = prime_factors(M);
    // dbg!(&m);
    let mut v_max = 0;
    for v in m.values().cloned() {
        v_max = max(v_max, v);
    }
    let mc = ModComb::new(v_max+1+N-1, p);
    let mut res= 1;
    for v in m.values() {
        res *= mc.nHk(v+1,N-1);
    }
    res %= p;
    println!("{}", res);
}
/// x ^ n % m
pub fn modpow(x: usize, n: usize, m: usize) -> usize {
    let mut res = 1;
    let mut x = x % m;
    let mut n = n;
    while n > 0 {
        if n & 1 == 1 {
            res = (res * x) % m;
        }
        x = (x * x) % m;
        n >>= 1;
    }
    res
}
struct ModComb {
    po: Vec<usize>,
    inv: Vec<usize>,
    n: usize,
    p: usize,
}
impl ModComb {
    // O(N)
    fn initialize(ft: &mut Self) {
        let n = ft.n as usize;

        ft.po[0] = 1;
        for i in 1..n {
            ft.po[i] = (ft.po[i-1] * i) % ft.p;
        }
        ft.inv[n-1] = modpow(ft.po[n-1], ft.p-2, ft.p);
        for i in (0..n-1).rev() {
            ft.inv[i] = (ft.inv[i+1] * (i+1)) % ft.p;
        }
    }
    fn new(max_n: usize, p: usize) -> ModComb {
        // nHk needs twice the length of the max_n.
        // since it doesn't affect the computational order, we always double the number for convenience.
        let mut ft = ModComb {
            po: vec![0; max_n+1 as usize],
            inv: vec![0; max_n+1 as usize],
            n: max_n+1,
            p: p,
        };
        Self::initialize(&mut ft);
        ft
    }
    fn nCk(&self, n: usize, k: usize) -> usize {
        if n < k { return 0; }
        (self.nPk(n,k) * self.inv[k as usize]) % self.p 
    }
    fn nPk(&self, n: usize, k: usize) -> usize {
        if n < k { return 0; }
        self.po[n as usize] * self.inv[(n-k) as usize] % self.p
    }
    fn nHk(&self, n: usize, k: usize) -> usize {
        if n==0 && k==0 { return 1 }
        self.nCk(n+k-1,k)
    }
}