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 Python (3.4.3) (judge verdict: Wrong Answer)

import sys

def sieve(n):
    """Sieve of Eratosthenes"""
    if n < 2:
        is_prime = [False for _ in range(n+1)]
        return is_prime

    is_prime = [True for _ in range(n+1)]
    is_prime[0] = False
    is_prime[1] = False

    primes = []
    for i in range(2, int(n**0.5)+1):
        if is_prime[i]:
            primes.append(i)
            for j in range(i*2, n+1, i):
                is_prime[j] = False

    return primes

def factorization(n, primes):
    d = dict()
    temp = n
    for i in primes:
        if i > temp:
            break

        if temp % i == 0:
            cnt = 0
            while temp % i == 0:
                cnt += 1
                temp //= i
            d[i] = cnt
    
    if temp != 1:
        d[temp] = 1
    
    if len(d) == 0:
        d[n] = 1
    
    return d

def main():
    input = sys.stdin.readline
    A, B = map(int, input().split())
    primes = sieve(10**6)

    Af = factorization(A, primes)
    Bf = factorization(B, primes)
    cnt = 1
    for a in Af:
        if a in Bf:
            cnt += 1

    return cnt

if __name__ == '__main__':
    print(main())