Ex - Distinct Multiples

AtCoder

IDabc236_h

Time2000ms

Memory256MB

Difficulty—

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: $N$ $M$ $D_1$ $\ldots$ $D_N$ [samples]

Samples

Input #1

3 7
2 3 4

Output #1

3

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

Input #2

3 3
1 2 2

Output #2

0

No sequence $A$ satisfies the conditions.

Input #3

6 1000000000000000000
380214083 420492929 929717250 666796775 209977152 770361643

Output #3

325683519

Be sure to find the count modulo $998244353$.

API Response (JSON)

{
  "problem": {
    "name": "Ex - Distinct Multiples",
    "description": {
      "content": "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 ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc236_h"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given are positive integers $N$, $M$, and a sequence of positive integers $D = (D_1, \\dots, D_N)$.\nFind the number of sequences of positive integers $A = (A_1, \\dots, A_N)$ that satisfy the following ...",
      "is_translate": false,
      "language": "English"
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}

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