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 LuaJIT (2.0.4) (judge verdict: Time Limit Exceeded)

local mod = 1000000007
local mfl = math.floor
local mmi = math.min

local function bmul(x, y)
  local x0, y0 = x % 31623, y % 31623
  local x1, y1 = mfl(x / 31623), mfl(y / 31623)
  return (x1 * y1 * 14122 + (x1 * y0 + x0 * y1) * 31623 + x0 * y0) % mod
end
local function badd(x, y)
  return (x + y) % mod
end

local function modpow(src, pow)
  local res = 1
  while 0 < pow do
    if pow % 2 == 1 then
      res = bmul(res, src)
      pow = pow - 1
    end
    src = bmul(src, src)
    pow = mfl(pow / 2)
  end
  return res
end

local function modinv(src)
  return modpow(src, mod - 2)
end

local n, x = io.read("*n", "*n")
local fact = {1}
for i = 2, 200 do
  fact[i] = bmul(fact[i - 1], i)
end
local combfact = {}
for i = 1, 200 do
  combfact[i] = {}
  local c = 1
  for j = 1, i do
    c = bmul(c, i + 1 - j)
    c = bmul(c, modinv(j))
    combfact[i][j] = bmul(c, fact[j])
  end
end
local function getCombFact(n, k)
  if n == 0 then return 1 end
  if k == 0 then return 1 end
  return combfact[n][k]
end

local a = {}
for i = 1, n do
  a[i] = io.read("*n")
end
table.sort(a)
local t = {}
for i = 1, n do
  t[i] = {}
end
for i = 1, n - 1 do
  t[i][x % a[i]] = getCombFact(n - 1, n - i)
end
t[n][x % a[n]] = 1

for src = n, 1, -1 do
  for val, cnt in pairs(t[src]) do
    for dst = src - 1, 1, -1 do
      local k = cnt
      local nval = val % a[dst]
      local step = src - dst - 1
      k = bmul(k, getCombFact(src - 2, step))
      local precnt = t[dst][nval]
      if precnt then
        t[dst][nval] = badd(precnt, k)
      else
        t[dst][nval] = k
      end
    end
  end
end
local ret = 0
for k, v in pairs(t[1]) do
  ret = badd(ret, bmul(k, v))
end
print(ret)