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: Accepted)

pub trait FromFragments: Sized {
    fn fragments_count() -> usize;
    fn from_fragments(fragments: &[&str]) -> Result<Self, String>;
}
impl FromFragments for () {
    fn fragments_count() -> usize {
        1
    }
    fn from_fragments(_fragments: &[&str]) -> Result<(), String> {
        Ok(())
    }
}
impl FromFragments for String {
    fn fragments_count() -> usize {
        1
    }
    fn from_fragments(fragments: &[&str]) -> Result<String, String> {
        Ok(fragments[0].to_string())
    }
}
macro_rules ! impl_from_fragments { ( $ ( $ t : ty ) * ) => { $ ( impl FromFragments for $ t { fn fragments_count ( ) -> usize { 1 } fn from_fragments ( fragments : & [ & str ] ) -> Result <$ t , String > { use std :: str :: FromStr ; <$ t >:: from_str ( fragments [ 0 ] ) . map_err ( | _ | { format ! ( "cannot parse \"{}\" as {}" , fragments [ 0 ] , stringify ! ( $ t ) ) } ) } } ) * } ; }
impl_from_fragments ! ( isize usize i8 u8 i16 u16 i32 u32 i64 u64 f32 f64 ) ;
pub trait FromLine: Sized {
    fn from_line(line: &str) -> Result<Self, String>;
}
fn split_into_fragments(line: &str) -> Vec<&str> {
    #[allow(deprecated)]
    line.trim_right_matches('\n').split_whitespace().collect()
}
impl<T: FromFragments> FromLine for T {
    fn from_line(line: &str) -> Result<T, String> {
        let fragments = split_into_fragments(line);
        if fragments.len() != T::fragments_count() {
            return Err(format!(
                "line \"{}\" has {} fragments, expected {}",
                line,
                fragments.len(),
                T::fragments_count()
            ));
        }
        T::from_fragments(&fragments)
    }
}
impl<T: FromFragments> FromLine for Vec<T> {
    fn from_line(line: &str) -> Result<Vec<T>, String> {
        let n = T::fragments_count();
        let fragments = split_into_fragments(line);
        if fragments.len() % n != 0 {
            return Err(format!(
                "line \"{}\" has {} fragments, expected multiple of {}",
                line,
                fragments.len(),
                n
            ));
        }
        let mut result = Vec::new();
        for chunk in fragments.chunks(n) {
            match T::from_fragments(chunk) {
                Ok(v) => result.push(v),
                Err(msg) => {
                    let flagment_msg = if n == 1 {
                        format!("fragment {}", result.len())
                    } else {
                        let l = result.len();
                        format!("fragments {}-{}", n * l + 1, (n + 1) * l)
                    };
                    return Err(format!("{} of line \"{}\": {}", flagment_msg, line, msg));
                }
            }
        }
        Ok(result)
    }
}
macro_rules ! impl_from_fragments_for_tuples { ( $ t : ident $ var : ident $ count : expr ) => ( ) ; ( $ t : ident $ var : ident $ count : expr ; $ ( $ inner_t : ident $ inner_var : ident $ inner_count : expr ) ;* ) => { impl_from_fragments_for_tuples ! ( $ ( $ inner_t $ inner_var $ inner_count ) ;* ) ; impl <$ t : FromFragments , $ ( $ inner_t : FromFragments ) ,*> FromFragments for ( $ t , $ ( $ inner_t ) ,* ) { fn fragments_count ( ) -> usize { let mut n = <$ t >:: fragments_count ( ) ; $ ( n += <$ inner_t >:: fragments_count ( ) ; ) * n } # [ allow ( unused_assignments ) ] fn from_fragments ( fragments : & [ & str ] ) -> Result < ( $ t , $ ( $ inner_t ) ,* ) , String > { let mut start = 0 ; let $ var = <$ t >:: from_fragments ( & fragments [ start .. start +<$ t >:: fragments_count ( ) ] ) ?; start += <$ t >:: fragments_count ( ) ; $ ( let $ inner_var = <$ inner_t >:: from_fragments ( & fragments [ start .. start +<$ inner_t >:: fragments_count ( ) ] ) ?; start += <$ inner_t >:: fragments_count ( ) ; ) * Ok ( ( $ var , $ ( $ inner_var ) ,* ) ) } } } }
impl_from_fragments_for_tuples ! ( T4 x4 4 ; T3 x3 3 ; T2 x2 2 ; T1 x1 1 ) ;
#[macro_export]
macro_rules ! read { ( ) => { let mut line = String :: new ( ) ; std :: io :: stdin ( ) . read_line ( & mut line ) . unwrap ( ) ; } ; ( $ pat : pat = $ t : ty ) => { let mut line = String :: new ( ) ; std :: io :: stdin ( ) . read_line ( & mut line ) . unwrap ( ) ; let $ pat = <$ t >:: from_line ( & line ) . unwrap ( ) ; } ; ( $ ( $ pat : pat = $ t : ty ) ,+ ) => { read ! ( ( $ ( $ pat ) ,* ) = ( $ ( $ t ) ,* ) ) ; } ; }
#[macro_export]
macro_rules ! readls { ( $ ( $ pat : pat = $ t : ty ) ,+ ) => { $ ( read ! ( $ pat = $ t ) ; ) * } }
pub fn read_vec<T: FromLine>() -> Vec<T> {
    use std::io::{self, BufRead};
    let stdin = io::stdin();
    let result = stdin
        .lock()
        .lines()
        .map(|line_result| {
            let line = line_result.expect("read from stdin failed");
            T::from_line(&line).unwrap()
        }).collect();
    result
}
#[macro_export]
macro_rules ! read_loop { ( $ pat : pat = $ t : ty ; $ block : block ) => { use std :: io :: BufRead ; let stdin = std :: io :: stdin ( ) ; for line in stdin . lock ( ) . lines ( ) { let line = line . expect ( "read from stdin failed" ) ; let $ pat = <$ t >:: from_line ( & line ) . unwrap ( ) ; $ block } } ; ( $ ( $ pat : pat = $ t : ty ) ,* ; $ block : block ) => { read_loop ! ( ( $ ( $ pat ) ,* ) = ( $ ( $ t ) ,* ) ; $ block ) ; } }
#[macro_export]
macro_rules ! readn_loop { ( $ n : expr ; $ pat : pat = $ t : ty ; $ block : block ) => { use std :: io :: BufRead ; let stdin = std :: io :: stdin ( ) ; { let lock = stdin . lock ( ) ; for _ in 0 ..$ n { let line = String :: new ( ) ; lock . read_line ( & mut line ) . expect ( "read from stdin failed" ) ; let $ pat = <$ t >:: from_line ( & line ) . unwrap ( ) ; $ block } } } ; ( $ n : expr ; $ ( $ pat : pat = $ t : ty ) ,* ; $ block : block ) => { read_times_loop ! ( $ n ; ( $ ( $ pat ) ,* ) = ( $ ( $ t ) ,* ) ; $ block ) ; } }
trait ReadExt<T> {
    fn read(&self) -> T;
}
impl<'a, T: FromFragments> ReadExt<T> for [&'a str] {
    fn read(&self) -> T {
        T::from_fragments(self).unwrap()
    }
}
impl<'a, T: FromFragments> ReadExt<T> for &'a str {
    fn read(&self) -> T {
        T::from_fragments(&[self]).unwrap()
    }
}
#[macro_export]
macro_rules! readable {
    ( $ t : ty , $ fragments_count : expr , |$ fragments : ident | $ from_fragments : expr ) => {
        impl FromFragments for $t {
            fn fragments_count() -> usize {
                $fragments_count
            }
            fn from_fragments($fragments: &[&str]) -> Result<Self, String> {
                Ok($from_fragments)
            }
        }
    };
}

// 重複組み合わせ ... multi-choose, combinations with replacement/repetition
// nHrのHは、homogeneous polynomialが由来

const MOD: u64 = 1_000_000_007;

use std::collections::HashMap;

fn factorize(mut m: u64) -> HashMap<u64, u64> {
    let mut res = HashMap::new();
    while m > 1 {
        for i in 2.. {
            if m % i == 0 {
                let count = res.entry(i).or_insert(0);
                *count += 1;
                m /= i;
                break;
            }
        }
    }
    res
}

fn multi_choose(n: u64, c: u64) -> u64 {
    choose(n+c-1, n-1)
}

fn choose(n: u64, c: u64) -> u64 {
    let numer = (0..n+1).rev().take(c as usize).fold(1, |acc, x| acc * x % MOD);
    let denom = (1..c+1).fold(1, |acc, x| acc * x % MOD);
    numer * inv(denom) % MOD
}

fn inv(n: u64) -> u64 {
    pow(n, MOD-2)
}

fn pow(n: u64, x: u64) -> u64 {
    if x == 0 { 1 }
    else if x % 2 == 0 {
        let p = pow(n, x/2);
        p * p % MOD
    } else {
        n * pow(n, x-1) % MOD
    }
}

fn main() {
    read!(n = u64, m = u64);
    let counts: Vec<u64> = factorize(m).into_iter().map(|(_, count)| count).collect();
    let mut ans = 1;
    for c in counts {
        ans = ans * multi_choose(n, c) % MOD;
    }
    println!("{}", ans);
}