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 Python (3.4.3) (judge verdict: Time Limit Exceeded)

def main():
    mod = 10**9+7
    fact = [1, 1]
    for i in range(2, 300):
        fact.append(fact[-1]*i % mod)

    n, x = map(int, input().split())
    s = sorted(list(map(int, input().split())))[::-1]
    memo = [dict() for _ in [0]*(n+1)]
    memo[0][x] = fact[n]
    for i in range(n):
        inv = pow(n-i, mod-2, mod)
        for j, a in memo[i].items():
            m = a*inv % mod
            for k, l in enumerate(s[i:]):
                memo[k+i+1][j % l] = (memo[k+i+1].get(j % l, 0)+m) % mod
    ans = 0
    for i, j in memo[n].items():
        ans = (ans+i*j) % mod
    print(ans)


main()