CF1866B Battling with Numbers

On 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

The 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 $ .

An 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 $ .

In 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 $ .