Ex - Product Modulo 2

AtCoder

IDabc245_h

Time2000ms

Memory256MB

Difficulty—

Among the sequences of length $K$ consisting of integers, $A=(A_1, \ldots, A_K)$, how many satisfy all of the conditions below? Find the count modulo $998244353$. * $0\leq A_i \leq M-1$ for every $i(1\leq i\leq K)$. * $\displaystyle\prod_{i=1}^{K} A_i \equiv N \pmod M$. ## Constraints * $1 \leq K \leq 10^9$ * $0 \leq N \lt M \leq 10^{12}$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $K$ $N$ $M$ [samples]

Samples

Input #1

2 3 6

Output #1

5

The five sequences $A$ satisfying the conditions are $(1,3),(3,1),(3,3),(3,5),(5,3)$.

Input #2

10 0 2

Output #2

Input #3

1000000000 20220326 1000000000000

Output #3

561382653

Be sure to find the count modulo $998244353$.

API Response (JSON)

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