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Ex - Distinct Multiples

Score : $600$ points

Problem Statement

Given are positive integers $N$, $M$, and a sequence of positive integers $D = (D_1, \dots, D_N)$.

Find the number of sequences of positive integers $A = (A_1, \dots, A_N)$ that satisfy the following conditions, modulo $998244353$.

• $1 \leq A_i \leq M \, (1 \leq i \leq N)$
• $A_i \neq A_j \, (1 \leq i \lt j \leq N)$
• For each $i \, (1 \leq i \leq N)$, $A_i$ is a multiple of $D_i$.

Constraints

• $2 \leq N \leq 16$
• $1 \leq M \leq 10^{18}$
• $1 \leq D_i \leq M \, (1 \leq i \leq N)$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$D_1$ $\ldots$ $D_N$

Output

Print the answer.

3 7
2 3 4

3

The three sequences $A$ that satisfy the conditions are $(2, 3, 4), (2, 6, 4), (6, 3, 4)$.

3 3
1 2 2

0

No sequence $A$ satisfies the conditions.

6 1000000000000000000
380214083 420492929 929717250 666796775 209977152 770361643

325683519

Be sure to find the count modulo $998244353$.