Subsequence LCM

AtCoder

IDabc349_f

Time2000ms

Memory256MB

Difficulty—

You are given a sequence of positive integers $A=(A_1,A_2,\dots,A_N)$ of length $N$ and a positive integer $M$. Find the number, modulo $998244353$, of non-empty and not necessarily contiguous subsequences of $A$ such that the least common multiple (LCM) of the elements in the subsequence is $M$. Two subsequences are distinguished if they are taken from different positions in the sequence, even if they coincide as sequences. Also, the LCM of a sequence with a single element is that element itself. ## Constraints * $1 \leq N \leq 2 \times 10^5$ * $1 \leq M \leq 10^{16}$ * $1 \leq A_i \leq 10^{16}$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ $A_1$ $A_2$ $\ldots$ $A_N$ [samples]

Samples

Input #1

4 6
2 3 4 6

Output #1

5

The subsequences of $A$ whose elements have an LCM of $6$ are $(2,3),(2,3,6),(2,6),(3,6),(6)$; there are five of them.

Input #2

5 349
1 1 1 1 349

Output #2

16

Note that even if some subsequences coincide as sequences, they are distinguished if they are taken from different positions.

Input #3

16 720720
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Output #3

API Response (JSON)

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      "statement_type": "Markdown",
      "content": "You are given a sequence of positive integers $A=(A_1,A_2,\\dots,A_N)$ of length $N$ and a positive integer $M$. Find the number, modulo $998244353$, of non-empty and not necessarily contiguous subsequ...",
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