Problem:
Score : 600 points

Problem Statement

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}

Output

Print the answer.

Sample Input 1

2 19
3 7

Sample 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.

Sample Input 2

5 82
22 11 6 5 13

Sample Output 2

288

Sample Input 3

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

Sample Output 3

279669259

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

Solution in PyPy3 (2.4.0) (judge verdict: Accepted)

import sys
import math
input = sys.stdin.readline
MOD = 10**9 + 7

n, k = [int(item) for item in input().split()]
s = [int(item) for item in input().split()]
s.sort(reverse=True)
MAX_VAL = max(max(s), k)

dp = [[0] * (MAX_VAL + 1) for _ in range(n + 1)]
dp[0][k] = 1
for i, item in enumerate(s):
    for j in range(MAX_VAL + 1):
        if dp[i][j] == 0:
            continue
        # Apply item
        dp[i+1][j % item] += dp[i][j]
        dp[i+1][j % item] %= MOD
        # Put item afterwards
        dp[i+1][j] += dp[i][j] * (n - 1 - i)
        dp[i+1][j] %= MOD

ans = 0
for i, item in enumerate(dp[n]):
    ans += i * item
    ans %= MOD
print(ans)