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: Time Limit Exceeded)

use std::cmp::Ordering::{Equal, Greater, Less};
use std::ops::Index;
use std::slice;

/** A prime generator, using the Sieve of Eratosthenes.

Create with `let mut pset = PrimeSet::new()`, and then use `pset.iter()` to iterate over all primes.
**/
#[derive(Default)]
pub struct PrimeSet {
    lst: Vec<u64>,
}

/// An iterator over generated primes. Created by `PrimeSet::iter` or
/// `PrimeSet::generator`
pub struct PrimeSetIter<'a> {
    p: &'a mut PrimeSet,
    n: usize,
    expand: bool,
}

impl PrimeSet {
    /// A new prime generator, primed with 2 and 3
    pub fn new() -> PrimeSet {
        PrimeSet { lst: vec![2, 3] }
    }

    /// Finds one more prime, and adds it to the list
    pub fn expand(&mut self) {
        let mut l: u64 = self.lst[self.lst.len() - 1] + 2;
        let mut remainder = 0;
        loop {
            for &n in &self.lst {
                remainder = l % n;
                if remainder == 0 || n * n > l {
                    break;
                }
            }

            if remainder != 0 {
                self.lst.push(l);
                break;
            };

            l += 2;
        }
    }

    /// Number of primes found so far
    pub fn len(&self) -> usize {
        self.lst.len()
    }

    pub fn is_empty(&self) -> bool {
        false
    }

    /// Return all primes found so far as a slice
    pub fn list(&self) -> &[u64] {
        &self.lst[..]
    }

    /// Iterator over all primes not yet found
    pub fn generator(&mut self) -> PrimeSetIter {
        let myn = self.len();
        PrimeSetIter {
            p: self,
            n: myn,
            expand: true,
        }
    }

    /// Iterator over all primes, starting with 2. If you don't care about the "state" of the
    /// `PrimeSet`, this is what you want!
    pub fn iter(&mut self) -> PrimeSetIter {
        PrimeSetIter {
            p: self,
            n: 0,
            expand: true,
        }
    }

    /// Iterator over just the primes found so far
    pub fn iter_vec(&self) -> slice::Iter<u64> {
        self.lst.iter()
    }

    /// Find the next largest prime from a number
    ///
    /// Returns `(idx, prime)`
    ///
    /// Note that if `n` is prime, then the output will be `(idx, n)`
    pub fn find(&mut self, n: u64) -> (usize, u64) {
        while n > *(self.lst.last().unwrap_or(&0)) {
            self.expand();
        }
        self.find_vec(n).unwrap()
    }

    /// Check if a number is prime
    ///
    /// Note that this only requires primes up to `n.sqrt()` to be generated, and will generate
    /// them as necessary on its own.
    #[cfg_attr(feature = "cargo-clippy", allow(wrong_self_convention))]
    pub fn is_prime(&mut self, n: u64) -> bool {
        if n <= 1 {
            return false;
        }
        if n == 2 {
            return true;
        } // otherwise we get 2 % 2 == 0!
        for m in self.iter() {
            if n % m == 0 {
                return false;
            };
            if m * m > n {
                return true;
            };
        }
        unreachable!("This iterator should not be empty.");
    }

    /// Find the next largest prime from a number, if it is within the already-found list
    ///
    /// Returns `(idx, prime)`
    ///
    /// Note that if `n` is prime, then the output will be `(idx, n)`
    pub fn find_vec(&self, n: u64) -> Option<(usize, u64)> {
        if n > *(self.lst.last().unwrap_or(&0)) {
            return None;
        }

        let mut base: usize = 0;
        let mut lim: usize = self.len();

        // Binary search algorithm
        while lim != 0 {
            let ix = base + (lim >> 1);
            match self.lst[ix].cmp(&n) {
                Equal => return Some((ix, self.lst[ix])),
                Less => {
                    base = ix + 1;
                    lim -= 1;
                }
                Greater => (),
            }
            lim >>= 1;
        }
        Some((base, self.lst[base]))
    }

    /// Get the nth prime, even if we haven't yet found it
    pub fn get(&mut self, index: usize) -> u64 {
        for _ in 0..(index as isize) + 1 - (self.lst.len() as isize) {
            self.expand();
        }
        self.lst[index]
    }

    /// Get the prime factors of a number, starting from 2, including repeats
    pub fn prime_factors(&mut self, n: u64) -> Vec<u64> {
        if n == 1 {
            return Vec::new();
        }
        let mut curn = n;
        let mut lst: Vec<u64> = Vec::new();
        for p in self.iter() {
            while curn % p == 0 {
                println!("Pushing {} ({} / {})", p, curn, n);
                lst.push(p);
                curn /= p;
                if curn == 1 {
                    return lst;
                }
            }

            if p * p > curn {
                println!("Final push {} ({} / {})", p, curn, n);
                lst.push(curn);
                return lst;
            }
        }
        unreachable!("This should be unreachable.");
    }
}

impl Index<usize> for PrimeSet {
    type Output = u64;
    fn index(&self, index: usize) -> &u64 {
        &self.lst[index]
    }
}

impl<'a> Iterator for PrimeSetIter<'a> {
    type Item = u64;
    fn next(&mut self) -> Option<u64> {
        while self.n >= self.p.len() {
            if self.expand {
                self.p.expand()
            } else {
                return None;
            }
        }
        self.n += 1;

        let m = self.p.lst[self.n - 1];

        Some(m)
    }
}

/// Find the first factor (other than 1) of a number
fn firstfac(x: u64) -> u64 {
    if x % 2 == 0 {
        return 2;
    };
    // TODO: return to step_by
    // for n in (3..).step_by(2).take_while(|m| m*m <= x) {
    for n in (1..).map(|m| 2 * m + 1).take_while(|m| m * m <= x) {
        if x % n == 0 {
            return n;
        };
    }
    // No factor found. It must be prime.
    x
}

/// Find all prime factors of a number
/// Does not use a `PrimeSet`, but simply counts upwards
pub fn factors(x: u64) -> Vec<u64> {
    if x <= 1 {
        return vec![];
    };
    let mut lst: Vec<u64> = Vec::new();
    let mut curn = x;
    loop {
        let m = firstfac(curn);
        lst.push(m);
        if m == curn {
            break;
        } else {
            curn /= m
        };
    }
    lst
}

/// Find all unique prime factors of a number
pub fn factors_uniq(x: u64) -> Vec<u64> {
    if x <= 1 {
        return vec![];
    };
    let mut lst: Vec<u64> = Vec::new();
    let mut curn = x;
    loop {
        let m = firstfac(curn);
        lst.push(m);
        if curn == m {
            break;
        }
        while curn % m == 0 {
            curn /= m;
        }
        if curn == 1 {
            break;
        }
    }
    lst
}

/// Test whether a number is prime. Checks every odd number up to `sqrt(n)`.
pub fn is_prime(n: u64) -> bool {
    if n <= 1 {
        return false;
    }
    firstfac(n) == n
}

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)*}
    };
}

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<_>>()
    };

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

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

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

fn gcd(a: usize, b: usize) -> usize {
    assert!(a != 0 || b != 0);
    if a == 0 { b }
    else { gcd(b % a, a) }
}

fn main() {
    input! {
        a: usize,
        b: usize,
    };

    let mut pset = PrimeSet::new();

    let mut ans = 1;
    for (_, n) in pset.iter().enumerate().take(gcd(a, b)) {
        if (a as u64) % n == 0 && (b as u64) % n == 0 {
            ans += 1;
        }
    }
    println!("{}", ans);
}