Problem:
Score : 400 points

Problem Statement

Given are positive integers A and B.

Let us choose some number of positive common divisors of A and B.

Here, any two of the chosen divisors must be coprime.

At most, how many divisors can we choose?

Definition of common divisor

An integer d is said to be a common divisor of integers x and y when d divides both x and y.

Definition of being coprime

Integers x and y are said to be coprime when x and y have no positive common divisors other than 1.

Definition of dividing

An integer x is said to divide another integer y when there exists an integer \alpha such that y = \alpha x.

Constraints

All values in input are integers.

1 \leq A, B \leq 10^{12}

Input

Input is given from Standard Input in the following format:

A B

Output

Print the maximum number of divisors that can be chosen to satisfy the condition.

Sample Input 1

12 18

Sample Output 1

3

12 and 18 have the following positive common divisors: 1, 2, 3, and 6.

1 and 2 are coprime, 2 and 3 are coprime, and 3 and 1 are coprime, so we can choose 1, 2, and 3, which achieve the maximum result.

Sample Input 2

420 660

Sample Output 2

4

Sample Input 3

1 2019

Sample Output 3

1

1 and 2019 have no positive common divisors other than 1.

Solution in Rust (1.15.1) (judge verdict: Accepted)

#![allow(unused_imports)]
#![allow(dead_code)]
use std::io::prelude::*;
use std::cmp::{max, min};
use std::collections::BinaryHeap;

const MOD: i64 = 1e9 as i64 + 7;

fn eratosthenes(p: &mut Vec<i64>, m: usize){
    let mut f: Vec<bool> = vec![true; m + 1];
    f[0] = false;
    f[1] = false;
    for i in 2..m + 1{
        if f[i]{
            p.push(i as i64);
        }
        let mut j = i * 2;
        while j <= m{
            f[j] = false;
            j += i;
        }
    }
}

fn is_prime(p: &Vec<i64>, x: i64) -> bool{
    for i in p.iter(){
        if *i >= x{
            break;
        }else if x % i == 0{
            return false;
        }
    }
    true
}

fn gcd(a: i64, b: i64) -> i64{
    if b == 0{
        a
    }else{
        gcd(b, a % b)
    }
}

fn main(){
    let a: i64 = read();
    let b: i64 = read();
    let mut p: Vec<i64> = vec![];
    eratosthenes(&mut p, 1e6 as usize);
    let g = gcd(a, b);
    let mut ans = 0;
    let mut i = 1;
    while i * i <= g{
        if g % i == 0{
            if is_prime(&p, i){ans += 1;}
            if i * i != g && is_prime(&p, g / i){ans += 1;}
        }
        i += 1;
    }
    println!("{}", ans);
}

fn read<T>() -> T
where
    T: std::str::FromStr,
{
    let stdin = std::io::stdin();
    let token: String = stdin
        .lock()
        .bytes()
        .map(|c| c.unwrap() as char)
        .skip_while(|c| c.is_whitespace())
        .take_while(|c| !c.is_whitespace())
        .collect();
    token.parse().ok().unwrap()
}