B. Battling with Numberstime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputOn the trip to campus during the mid semester exam period, Chaneka thinks of two positive integers X and Y. Since the two integers can be very big, both are represented using their prime factorisations, such that:   X=A_1^{B_1}\times A_2^{B_2}\times\ldots\times A_N^{B_N} (each A_i is prime, each B_i is positive, and A_1<A_2<\ldots<A_N)  Y=C_1^{D_1}\times C_2^{D_2}\times\ldots\times C_M^{D_M} (each C_j is prime, each D_j is positive, and C_1<C_2<\ldots<C_M) Chaneka ponders about these two integers for too long throughout the trip, so Chaneka's friend commands her "Gece, deh!" (move fast) in order to not be late for the exam.Because of that command, Chaneka comes up with a problem, how many pairs of positive integers p and q such that \text{LCM}(p, q) = X and \text{GCD}(p, q) = Y. Since the answer can be very big, output the answer modulo 998\,244\,353.Notes:   \text{LCM}(p, q) is the smallest positive integer that is simultaneously divisible by p and q.  \text{GCD}(p, q) is the biggest positive integer that simultaneously divides p and q. InputThe first line contains a single integer N (1 \leq N \leq 10^5) — the number of distinct primes in the prime factorisation of X.The second line contains N integers A_1, A_2, A_3, \ldots, A_N (2 \leq A_1 < A_2 < \ldots < A_N \leq 2 \cdot 10^6; each A_i is prime) — the primes in the prime factorisation of X.The third line contains N integers B_1, B_2, B_3, \ldots, B_N (1 \leq B_i \leq 10^5) — the exponents in the prime factorisation of X.The fourth line contains a single integer M (1 \leq M \leq 10^5) — the number of distinct primes in the prime factorisation of Y.The fifth line contains M integers C_1, C_2, C_3, \ldots, C_M (2 \leq C_1 < C_2 < \ldots < C_M \leq 2 \cdot 10^6; each C_j is prime) — the primes in the prime factorisation of Y.The sixth line contains M integers D_1, D_2, D_3, \ldots, D_M (1 \leq D_j \leq 10^5) — the exponents in the prime factorisation of Y.OutputAn integer representing the number of pairs of positive integers p and q such that \text{LCM}(p, q) = X and \text{GCD}(p, q) = Y, modulo 998\,244\,353.ExamplesInput
4
2 3 5 7
2 1 1 2
2
3 7
1 1
Output
8
Input
2
1299721 1999993
100000 265
2
1299721 1999993
100000 265
Output
1
Input
2
2 5
1 1
2
2 3
1 1
Output
0
NoteIn the first example, the integers are as follows:   X=2^2\times3^1\times5^1\times7^2=2940  Y=3^1\times7^1=21 The following are all possible pairs of p and q:   p=21, q=2940  p=84, q=735  p=105, q=588  p=147, q=420  p=420, q=147  p=588, q=105  p=735, q=84  p=2940, q=21 In the third example, the integers are as follows:   X=2^1\times5^1=10  Y=2^1\times3^1=6 There is no pair p and q that simultaneously satisfies \text{LCM}(p,q)=10 and \text{GCD}(p,q)=6.
4
2 3 5 7
2 1 1 2
2
3 7
1 1

8 
['combinatorics', 'math', 'number theory', '*1400']