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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Linq;
using System.Collections.Generic;
using System.Text;

using SyrupBeans.Competitive.Integer;

using E = System.Linq.Enumerable;

public class Program
{
    public static void Main()
    {
        var ab = Console.ReadLine().Split().Select(x => long.Parse(x)).ToArray();
        var a = ab[0];
        var b = ab[1];

        var commonDivisors = new HashSet<long>(Divisibility.EnumerateDivisors(a));
        commonDivisors.IntersectWith(new HashSet<long>(Divisibility.EnumerateDivisors(b)));

        var k = 1;
        var candidates = new SortedSet<long>(commonDivisors);
        candidates.Remove(1);
        while (candidates.Any())
        {
            ++k;
            var t = candidates.Min + 1;
            while (t <= candidates.Max)
            {
                var view = candidates.GetViewBetween(t, candidates.Max);
                t = view.Min + 1;
                if (view.Min % candidates.Min != 0)
                {
                    continue;
                }
                candidates.Remove(view.Min);
            }
            candidates.Remove(candidates.Min);
        }

        Console.WriteLine(k);
    }
}

namespace SyrupBeans.Competitive.Integer
{
    using Integer = System.Int64;
    using System;
    using System.Collections.Generic;
    using System.Linq;

    public static class Divisibility
    {
        
        /// <summary>O(log max(<paramref name="x" />, <paramref name="y" />))</summary>
        public static Integer Gcd(Integer x, Integer y) => x < 1 ? y : Gcd(y % x, x);

        /// <summary>O(sqrt(<paramref name="x" />))</summary>
        public static IEnumerable<Integer> EnumerateDivisors(Integer x)
        {
            var cache = new Stack<Integer>();
            for (Integer i = 1; i * i < x; ++i)
            {
                Integer r, d = Math.DivRem(x, i, out r);
                if (r != 0)
                {
                    continue;
                }
                yield return i;
                cache.Push(d);
            }
            if (cache.Any() && cache.Peek() * cache.Peek() == x)
            {
                cache.Pop();
            }
            while (cache.Any())
            {
                yield return cache.Pop();
            }
        }
    }
}