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: Time Limit Exceeded)

import math
from functools import lru_cache

@lru_cache(maxsize=10000)
def fact(n):
    return math.factorial(n) % (10 ** 9 + 7)


N, X = [int(x) for x in input().split()]
S = [int(x) for x in input().split()]

tab = [0] * min(S)

S.sort()

@lru_cache(maxsize=10000)
def f(NUM, LIST):
    modL = [NUM % l for l in LIST]
    ret = 0
    for i in range(len(LIST)):
        l = modL[i]

        if l > min(S):
            newL = list(LIST)
            del newL[i]
            ret += f(l, tuple(newL))
        elif l == min(S):
            pass
        else:
            ret += l * fact(len(LIST) - 1)

    return ret % (10 ** 9 + 7)


print(f(X, tuple(S)) )