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.



Sample input:
12 18


Sample output:
3

Submission status: Time Limit Exceeded

Solution code:
import math

a, b = [int(i) for i in input().split()]


class Prime:
    seed_primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
    seed_primes = [int(i) for i in "2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 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36677 36683 36691 36697 36709 36713 36721 36739 36749 36761 36767 36779 36781 36787 36791 36793 36809 36821 36833 36847 36857 36871 36877 36887 36899 36901 36913 36919 36923 36929 36931 36943 36947 36973 36979 36997 37003 37013 37019 37021 37039 37049 37057 37061 37087 37097 37117 37123 37139 37159 37171 37181 37189 37199 37201 37217 37223 37243 37253 37273 37277 37307 37309 37313 37321 37337 37339 37357 37361 37363 37369 37379 37397 37409 37423 37441 37447 37463 37483 37489 37493 37501 37507 37511 37517 37529 37537 37547 37549 37561 37567 37571 37573 37579 37589 37591 37607 37619 37633 37643 37649 37657 37663 37691 37693 37699 37717 37747 37781 37783 37799 37811 37813 37831 37847 37853 37861 37871 37879 37889 37897 37907 37951 37957 37963 37967 37987 37991 37993 37997 38011 38039 38047 38053 38069 38083 38113 38119 38149 38153 38167 38177 38183 38189 38197 38201 38219 38231 38237 38239 38261 38273 38281 38287 38299 38303 38317 38321 38327 38329 38333 38351 38371 38377 38393 38431 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40169 40177 40189 40193 40213 40231 40237 40241 40253 40277 40283 40289 40343 40351 40357 40361 40387 40423 40427 40429 40433 40459 40471 40483 40487 40493 40499 40507 40519 40529 40531 40543 40559 40577 40583 40591 40597 40609 40627 40637 40639 40693 40697 40699 40709 40739 40751 40759 40763 40771 40787 40801 40813 40819 40823 40829 40841 40847 40849 40853 40867 40879 40883 40897 40903 40927 40933 40939 40949 40961 40973 40993 41011 41017 41023 41039 41047 41051 41057 41077 41081 41113 41117 41131 41141 41143 41149 41161 41177 41179 41183 41189 41201 41203 41213 41221 41227 41231 41233 41243 41257 41263 41269 41281 41299 41333 41341 41351 41357 41381 41387 41389 41399 41411 41413 41443 41453 41467 41479 41491 41507 41513 41519 41521 41539 41543 41549 41579 41593 41597 41603 41609 41611 41617 41621 41627 41641 41647 41651 41659 41669 41681 41687 41719 41729 41737 41759 41761 41771 41777 41801 41809 41813 41843 41849 41851 41863 41879 41887 41893 41897 41903 41911 41927 41941 41947 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    def is_prime(self, n):
        """
        prime test (hybrid)

        see also: https://qiita.com/gushwell/items/ff9ed83ba55350aaa369

        :param n:
        :return: boolean
        """
        is_prime_common = self.is_prime_common(n)
        if is_prime_common is not None:
            return is_prime_common

        if n < 2000000:
            return self.is_prime_brute_force(n)
        else:
            return self.is_prime_miller_rabin(n)

    def is_prime_common(self, n):
        if n == 1: return False
        if n in Prime.seed_primes: return True
        if any(map(lambda x: n % x == 0, self.seed_primes)): return False

    def is_prime_brute_force(self, n):
        """
        brute force prime test
        use with is_prime_common if you want to skip seed_primes

        :param n:
        :return: boolean
        """
        for k in range(2, int(math.sqrt(n)) + 1):
            if n % k == 0:
                return False
        return True

    def is_prime_miller_rabin(self, n):
        """
        miller rabin prime test
        use with is_prime_common if you want to skip seed_primes

        see also
            algorithm: https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test
            implementation: https://qiita.com/srtk86/items/609737d50c9ef5f5dc59
            improvement: https://qiita.com/gushwell/items/ff9ed83ba55350aaa369

        :param n:
        :return: boolean
        """

        d = n - 1
        while d & 1 == 0:
            d >>= 1

        # use one of these lines / upper is more efficient.
        witnesses = self.get_witnesses(n)
        # witnesses = [random.randint(1, n - 1) for _ in range(100)]

        for w in witnesses:
            y = pow(w, d, n)

            while d != n - 1 and y != 1 and y != n - 1:
                y = (y * y) % n
                d <<= 1

            if y != n - 1 and d & 1 == 0:
                return False

        return True

    def get_witnesses(self, num):
        def _get_range(num):
            if num < 2047:
                return 1
            if num < 1373653:
                return 2
            if num < 25326001:
                return 3
            if num < 3215031751:
                return 4
            if num < 2152302898747:
                return 5
            if num < 3474749660383:
                return 6
            if num < 341550071728321:
                return 7
            if num < 3825123056546413051:
                return 9
            return 12

        return self.seed_primes[:_get_range(num)]

    def gcd(self, a, b):
        if a < b:
            return self.gcd(b, a)
        if b == 0:
            return a
        while b:
            a, b = b, a % b
        return a

    @staticmethod
    def f(x, n, seed):
        """
        pseudo prime generator
        :param x:
        :param n:
        :param seed:
        :return: pseudo prime
        """
        p = Prime.seed_primes[seed % len(Prime.seed_primes)]
        return (p * x + seed) % n

    def find_factor(self, n, seed=1):
        """
        find one of factor of n
        this function is based to Pollard's rho algorithm

        see also
            algorithm: https://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm
            implementation: https://qiita.com/gushwell/items/561afde2e00bf3380c98

        :param n:
        :param seed:
        :return: factor
        """
        if self.is_prime(n):
            return n

        x, y, d = 2, 2, 1
        count = 0
        while d == 1:
            count += 1
            x = self.f(x, n, seed)
            y = self.f(self.f(y, n, seed), n, seed)
            d = self.gcd(abs(x - y), n)

        if d == n:
            return self.find_factor(n, seed + 1)
        return self.find_factor(d)

    def find_factors(self, n):
        primes = {}
        if self.is_prime(n):
            primes[n] = 1
            return primes

        while n > 1:
            factor = self.find_factor(n)

            primes.setdefault(factor, 0)
            primes[factor] += 1

            n //= factor

        return primes


prime = Prime()
_gcd = prime.gcd(a, b)
fact_dict = prime.find_factors(_gcd)
print(len(fact_dict.keys()) + 1)