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)

[N, X] = list(map(int, input().split()))
s_list = sorted(list(map(int, input().split())))

answer = 0
mod = 10 ** 9 + 7
memo_dict = {}
fact_dict = {}
for i in range(N):
    if i == 0:
        fact_dict[i] = 1
    else:
        fact_dict[i] = fact_dict[i - 1] * i % mod

# print(fact_dict)


def sumD(x, s, s_list):
    s_list_tmp = s_list[:]
    next_mod = x % s
    s_list_tmp.extend([next_mod, s])
    key = tuple(s_list_tmp)
    if key in memo_dict:
        # print("hit")
        return memo_dict[key]
    min_s = min(s_list)

    if next_mod < min_s:
        value = fact_dict[len(s_list) - 1] * next_mod % mod
        # print("register")
        # print(key, value)
        memo_dict[key] = value
        return value

    if s == min_s:
        value = fact_dict[len(s_list) - 1] * next_mod % mod
        # print("register")
        # print(key, value)
        memo_dict[key] = value
        return value

    s_list2 = s_list[:]
    s_list2.remove(s)
    value = sum([sumD(next_mod, s_i, s_list2) % mod for s_i in s_list2])
    # print("register")
    # print(key, value)
    memo_dict[key] = value
    return value


answer = sum([sumD(X, s, s_list) % mod for s in s_list]) % mod

print(answer)