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 Common Lisp (SBCL 1.1.14) (judge verdict: Accepted)

(defun prime-factorization (n)
  (let ((ans '()))
    (labels ((divide (m i c)
		     (unless (zerop (rem m i))
		       (return-from divide (values m (cons i c))))
		     (divide (/ m i) i (1+ c)))
	     (take-apart (m i)
			 (when (= m 1)
			   (return-from take-apart ans))

			 (when (< (sqrt m) i)
			   (return-from take-apart
			     (setf ans (append ans (list (cons m 1))))))

			 (if (zerop (rem m i))
			    (multiple-value-bind (m2 part) (divide m i 0)
			      (setf ans (append ans (list part)))
			      (take-apart m2 (1+ i)))
			    (take-apart m (1+ i)))))
      (take-apart n 2)
      ans)))

(defun abc142d ()
  (let* ((a (read))
	 (b (read)))
    (format t "~a~%"
	    (1+
	     (length
	       (intersection
		 (mapcar #'car (prime-factorization a))
		 (mapcar #'car (prime-factorization b))))))))

(abc142d)