Modulo Operations

AtCoder

IDexawizards2019_d

Time2000ms

Memory256MB

Difficulty—

Snuke has a blackboard and a set $S$ consisting of $N$ integers. The $i$\-th element in $S$ is $S_i$. He wrote an integer $X$ on the blackboard, then performed the following operation $N$ times: * Choose one element from $S$ and remove it. * Let $x$ be the number written on the blackboard now, and $y$ be the integer removed from $S$. Replace the number on the blackboard with $x \bmod {y}$. There are $N!$ possible orders in which the elements are removed from $S$. For each of them, find the number that would be written on the blackboard after the $N$ operations, and compute the sum of all those $N!$ numbers modulo $10^{9}+7$. ## Constraints * All values in input are integers. * $1 \leq N \leq 200$ * $1 \leq S_i, X \leq 10^{5}$ * $S_i$ are pairwise distinct. ## Input Input is given from Standard Input in the following format: $N$ $X$ $S_1$ $S_2$ $\ldots$ $S_{N}$ [samples]

Samples

Input #1

2 19
3 7

Output #1

3

*   There are two possible orders in which we remove the numbers from $S$.
*   If we remove $3$ and $7$ in this order, the number on the blackboard changes as follows: $19 \rightarrow 1 \rightarrow 1$.
*   If we remove $7$ and $3$ in this order, the number on the blackboard changes as follows: $19 \rightarrow 5 \rightarrow 2$.
*   The output should be the sum of these: $3$.

Input #2

5 82
22 11 6 5 13

Output #2

Input #3

10 100000
50000 50001 50002 50003 50004 50005 50006 50007 50008 50009

Output #3

279669259

*   Be sure to compute the sum modulo $10^{9}+7$.

API Response (JSON)

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    "name": "Modulo Operations",
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      "content": "Snuke has a blackboard and a set $S$ consisting of $N$ integers. The $i$\\-th element in $S$ is $S_i$. He wrote an integer $X$ on the blackboard, then performed the following operation $N$ times: *   ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
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    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "exawizards2019_d"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Snuke has a blackboard and a set $S$ consisting of $N$ integers. The $i$\\-th element in $S$ is $S_i$.\nHe wrote an integer $X$ on the blackboard, then performed the following operation $N$ times:\n\n*   ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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