Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3

import sys
import math
# import re                                    # re.compile(pattern) => ptn obj; p.search(s), p.match(s), p.finditer(s) => match obj; p.sub(after, s)
# from collections import deque                # deque class. deque(L): dq.append(x), dq.appendleft(x), dq.pop(), dq.popleft(), dq.rotate()
# from collections import defaultdict          # subclass of dict. defaultdict(facroty)
# from collections import Counter              # subclass of dict. Counter(iter): c.elements(), c.most_common(n), c.subtract(iter)
# from heapq import heapify, heappush, heappop # built-in list. heapify(L) changes list in-place to min-heap in O(n), heappush(heapL, x) and heappop(heapL) in O(lgn).
# from heapq import nlargest, nsmallest        # nlargest(n, iter[, key]) returns k-largest-list in O(n+klgn).
# from itertools import count, cycle, repeat   # count(start[,step]), cycle(iter), repeat(elm[,n])
# from itertools import groupby                # [(k, list(g)) for k, g in groupby('000112')] returns [('0',['0','0','0']), ('1',['1','1']), ('2',['2'])]
# from itertools import starmap                # starmap(pow, [[2,5], [3,2]]) returns [32, 9]
# from itertools import product                # product(iter, repeat=n)
# from itertools import accumulate             # accumulate(iter[, f])
# from functools import reduce                 # reduce(f, iter[, init])
# from functools import lru_cache              # @lrucache ...arguments of functions should be able to be keys of dict
# from bisect import bisect_left, bisect_right # bisect_left(a, x, lo=0, hi=len(a)) returns i such that all(val<x for val in a[lo:i]) and all(val>-=x for val in a[i:hi]).
# from copy import deepcopy                    # to copy multi-dimentional matrix without reference


def main():
    mod = 1000000007                  # 10^9+7
    inf = float('inf')
    sys.setrecursionlimit(10**6)    # 1000 -> 1000000
    def input(): return sys.stdin.readline().rstrip()
    def ii():  return int(input())
    def mi():  return map(int, input().split())
    def mi_0(): return map(lambda x: int(x)-1, input().split())
    def lmi(): return list(map(int, input().split()))
    def lmi_0(): return list(map(lambda x: int(x)-1, input().split()))
    def li():  return list(input())
                

    from fractions import gcd
    a, b = mi()
    g = gcd(a, b)

    table = [True for _ in range(g+1)]
    n = int(math.sqrt(g) + 1)
    table[0] = False
    table[1] = False
    # 2 の倍数については特別扱いをして定数倍高速化
    for p in range(3, n, 2):
        if table[p]:
            for i in range(p**2, n, p):
                table[i] = False
    
    def calc_prime(num):
        cnt = 0
        # print(table)
        if num % 2 == 0:
            cnt += 1
            while num >= 2 and num % 2 == 0:
                num //= 2
        p = 3
        while p <= num:
            if table[p] and num % p == 0:
                cnt += 1
                while num >= p and num % p == 0:
                    num //= p
            p += 2
        return cnt

    print(calc_prime(g)+1)    # 1 分をたす

if __name__ == "__main__":
    main()

Fixed submission:
#!/usr/bin/env python3

import sys
import math
# import re                                    # re.compile(pattern) => ptn obj; p.search(s), p.match(s), p.finditer(s) => match obj; p.sub(after, s)
# from collections import deque                # deque class. deque(L): dq.append(x), dq.appendleft(x), dq.pop(), dq.popleft(), dq.rotate()
# from collections import defaultdict          # subclass of dict. defaultdict(facroty)
# from collections import Counter              # subclass of dict. Counter(iter): c.elements(), c.most_common(n), c.subtract(iter)
# from heapq import heapify, heappush, heappop # built-in list. heapify(L) changes list in-place to min-heap in O(n), heappush(heapL, x) and heappop(heapL) in O(lgn).
# from heapq import nlargest, nsmallest        # nlargest(n, iter[, key]) returns k-largest-list in O(n+klgn).
# from itertools import count, cycle, repeat   # count(start[,step]), cycle(iter), repeat(elm[,n])
# from itertools import groupby                # [(k, list(g)) for k, g in groupby('000112')] returns [('0',['0','0','0']), ('1',['1','1']), ('2',['2'])]
# from itertools import starmap                # starmap(pow, [[2,5], [3,2]]) returns [32, 9]
# from itertools import product                # product(iter, repeat=n)
# from itertools import accumulate             # accumulate(iter[, f])
# from functools import reduce                 # reduce(f, iter[, init])
# from functools import lru_cache              # @lrucache ...arguments of functions should be able to be keys of dict
# from bisect import bisect_left, bisect_right # bisect_left(a, x, lo=0, hi=len(a)) returns i such that all(val<x for val in a[lo:i]) and all(val>-=x for val in a[i:hi]).
# from copy import deepcopy                    # to copy multi-dimentional matrix without reference


def main():
    mod = 1000000007                  # 10^9+7
    inf = float('inf')
    sys.setrecursionlimit(10**6)    # 1000 -> 1000000
    def input(): return sys.stdin.readline().rstrip()
    def ii():  return int(input())
    def mi():  return map(int, input().split())
    def mi_0(): return map(lambda x: int(x)-1, input().split())
    def lmi(): return list(map(int, input().split()))
    def lmi_0(): return list(map(lambda x: int(x)-1, input().split()))
    def li():  return list(input())
                

    from fractions import gcd
    a, b = mi()
    g = gcd(a, b)
    # g = math.gcd(a, b)

    n = int(math.sqrt(g) + 1)
    table = [True for _ in range(n)]
    table[0] = False
    table[1] = False
    # 2 の倍数については特別扱いをして定数倍高速化
    primes = [2]
    for p in range(3, n, 2):
        if table[p]:
            primes.append(p)
            for i in range(p**2, n, p):
                table[i] = False
    
    def calc_prime(num):
        cnt = 0
        # print(table)
        # print(primes)
        for p in primes:
            if p > num:
                break
            if num % p == 0:
                cnt += 1
                while p <= num and num % p == 0:
                    num //= p
        if num != 1:
            cnt += 1    # sqrt(num) までわりまくってなおそれ以上の数が約数として残っている -> こいつも素数
        return cnt

    print(calc_prime(g)+1)    # 1 分をたす

if __name__ == "__main__":
    main()

Buggy python submission (verdict: Runtime Error):
#!/usr/bin/env python3

import sys
import math
# import re                                    # re.compile(pattern) => ptn obj; p.search(s), p.match(s), p.finditer(s) => match obj; p.sub(after, s)
# from collections import deque                # deque class. deque(L): dq.append(x), dq.appendleft(x), dq.pop(), dq.popleft(), dq.rotate()
# from collections import defaultdict          # subclass of dict. defaultdict(facroty)
# from collections import Counter              # subclass of dict. Counter(iter): c.elements(), c.most_common(n), c.subtract(iter)
# from heapq import heapify, heappush, heappop # built-in list. heapify(L) changes list in-place to min-heap in O(n), heappush(heapL, x) and heappop(heapL) in O(lgn).
# from heapq import nlargest, nsmallest        # nlargest(n, iter[, key]) returns k-largest-list in O(n+klgn).
# from itertools import count, cycle, repeat   # count(start[,step]), cycle(iter), repeat(elm[,n])
# from itertools import groupby                # [(k, list(g)) for k, g in groupby('000112')] returns [('0',['0','0','0']), ('1',['1','1']), ('2',['2'])]
# from itertools import starmap                # starmap(pow, [[2,5], [3,2]]) returns [32, 9]
# from itertools import product                # product(iter, repeat=n)
# from itertools import accumulate             # accumulate(iter[, f])
# from functools import reduce                 # reduce(f, iter[, init])
# from functools import lru_cache              # @lrucache ...arguments of functions should be able to be keys of dict
# from bisect import bisect_left, bisect_right # bisect_left(a, x, lo=0, hi=len(a)) returns i such that all(val<x for val in a[lo:i]) and all(val>-=x for val in a[i:hi]).
# from copy import deepcopy                    # to copy multi-dimentional matrix without reference


def main():
    mod = 1000000007                  # 10^9+7
    inf = float('inf')
    sys.setrecursionlimit(10**6)    # 1000 -> 1000000
    def input(): return sys.stdin.readline().rstrip()
    def ii():  return int(input())
    def mi():  return map(int, input().split())
    def mi_0(): return map(lambda x: int(x)-1, input().split())
    def lmi(): return list(map(int, input().split()))
    def lmi_0(): return list(map(lambda x: int(x)-1, input().split()))
    def li():  return list(input())
                

    from fractions import gcd
    a, b = mi()
    g = math.gcd(a, b)
    # g = gcd(a, b)

    table = [True for _ in range(g+1)]
    n = int(math.sqrt(g) + 1)
    table[0] = False
    table[1] = False
    # 2 の倍数については特別扱いをして定数倍高速化
    for p in range(3, n, 2):
        if table[p]:
            for i in range(p**2, n, p):
                table[i] = False
    
    def calc_prime(num):
        cnt = 0
        # print(table)
        if num % 2 == 0:
            cnt += 1
            while num >= 2 and num % 2 == 0:
                num //= 2
        p = 3
        while p <= num:
            if table[p] and num % p == 0:
                cnt += 1
                while num >= p and num % p == 0:
                    num //= p
            p += 2
        return cnt

    print(calc_prime(g)+1)    # 1 分をたす

if __name__ == "__main__":
    main()

Fixed submission:
#!/usr/bin/env python3

import sys
import math
# import re                                    # re.compile(pattern) => ptn obj; p.search(s), p.match(s), p.finditer(s) => match obj; p.sub(after, s)
# from collections import deque                # deque class. deque(L): dq.append(x), dq.appendleft(x), dq.pop(), dq.popleft(), dq.rotate()
# from collections import defaultdict          # subclass of dict. defaultdict(facroty)
# from collections import Counter              # subclass of dict. Counter(iter): c.elements(), c.most_common(n), c.subtract(iter)
# from heapq import heapify, heappush, heappop # built-in list. heapify(L) changes list in-place to min-heap in O(n), heappush(heapL, x) and heappop(heapL) in O(lgn).
# from heapq import nlargest, nsmallest        # nlargest(n, iter[, key]) returns k-largest-list in O(n+klgn).
# from itertools import count, cycle, repeat   # count(start[,step]), cycle(iter), repeat(elm[,n])
# from itertools import groupby                # [(k, list(g)) for k, g in groupby('000112')] returns [('0',['0','0','0']), ('1',['1','1']), ('2',['2'])]
# from itertools import starmap                # starmap(pow, [[2,5], [3,2]]) returns [32, 9]
# from itertools import product                # product(iter, repeat=n)
# from itertools import accumulate             # accumulate(iter[, f])
# from functools import reduce                 # reduce(f, iter[, init])
# from functools import lru_cache              # @lrucache ...arguments of functions should be able to be keys of dict
# from bisect import bisect_left, bisect_right # bisect_left(a, x, lo=0, hi=len(a)) returns i such that all(val<x for val in a[lo:i]) and all(val>-=x for val in a[i:hi]).
# from copy import deepcopy                    # to copy multi-dimentional matrix without reference


def main():
    mod = 1000000007                  # 10^9+7
    inf = float('inf')
    sys.setrecursionlimit(10**6)    # 1000 -> 1000000
    def input(): return sys.stdin.readline().rstrip()
    def ii():  return int(input())
    def mi():  return map(int, input().split())
    def mi_0(): return map(lambda x: int(x)-1, input().split())
    def lmi(): return list(map(int, input().split()))
    def lmi_0(): return list(map(lambda x: int(x)-1, input().split()))
    def li():  return list(input())
                

    from fractions import gcd
    a, b = mi()
    g = gcd(a, b)
    # g = math.gcd(a, b)

    n = int(math.sqrt(g) + 1)
    table = [True for _ in range(n)]
    table[0] = False
    table[1] = False
    # 2 の倍数については特別扱いをして定数倍高速化
    primes = [2]
    for p in range(3, n, 2):
        if table[p]:
            primes.append(p)
            for i in range(p**2, n, p):
                table[i] = False
    
    def calc_prime(num):
        cnt = 0
        # print(table)
        # print(primes)
        for p in primes:
            if p > num:
                break
            if num % p == 0:
                cnt += 1
                while p <= num and num % p == 0:
                    num //= p
        if num != 1:
            cnt += 1    # sqrt(num) までわりまくってなおそれ以上の数が約数として残っている -> こいつも素数
        return cnt

    print(calc_prime(g)+1)    # 1 分をたす

if __name__ == "__main__":
    main()