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 (2.7.6) (judge verdict: Runtime Error)

n, x = map(int, input().split())
s = list(map(int, input().split()))
s = sorted(s)
MOD = 10**9 + 7

frac = [0]*(n+1)
frac[0] = 1
for i in range(0,n):
    frac[i+1] = (i+1) * frac[i]
    frac[i+1] %= MOD

def solve(x, cnt, visited):
    ans = 0
    flag = False
    if cnt == n:
        return x
    for i in range(n):
        if not visited[i]:
            if s[i] <= x:
                if not flag:
                    ans += (x % s[i]) * frac[n - (cnt + 1)]
                    flag = True
                else:
                    tmp_visited = visited[0:]
                    tmp_visited[i] = True
                    ans += solve(x % s[i], cnt + 1, tmp_visited) % MOD
                    ans %= MOD
            else:
                tmp_visited = visited[0:]
                tmp_cnt = 0
                for j in range(i, n):
                    if not tmp_visited[j]:
                        tmp_visited[j] = True
                        tmp_cnt += 1
                ans += solve(x, cnt + tmp_cnt, tmp_visited) * frac[tmp_cnt] % MOD
                ans %= MOD
                break
    return ans
    
    
print(solve(x, 0, visited = [False]*n) % MOD)