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 Haskell (GHC 7.10.3) (judge verdict: Wrong Answer)

import qualified Data.Text as T
import qualified Data.Text.IO as T
import qualified Data.Text.Read as T

main :: IO ()
main = do
  (a : b : _) <- map unsafeSignedDecimal . T.words <$> T.getLine :: IO [Int]
  print . f $ gcd a b

f :: Int -> Int
f = \n -> fst . foldl step (1, n) $ [2..(n `div` 2)]
  where
    step a@(_, 1) _ = a
    step a@(k, m) i | r /= 0 = a
                    | otherwise = (succ k, g i q)
      where
        (q, r) = m `divMod` i
    g i = h
      where
        h m | r /= 0 = m
            | otherwise = h q
          where
            (q, r) = m `divMod` i

unsafeSignedDecimal :: T.Text -> Int
unsafeSignedDecimal s = case T.signed T.decimal s of
  Right (n, _) -> n