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 Lua (LuaJIT 2.1.0) (judge verdict: Accepted)

local mod = 1000000007
local mfl = math.floor

local function bmul(x, y)
  local x1, y1 = mfl(x / 31623), mfl(y / 31623)
  local x0, y0 = x - x1 * 31623, y - y1 * 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 bsub(x, y)
  return x < y and x - y + mod or x - y
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 fact = {1}
local invs = {1}
local invfact = {1}
for i = 2, 100010 do
  fact[i] = bmul(fact[i - 1], i)
  invs[i] = bmul(mfl(mod / i), mod - invs[mod % i])
  invfact[i] = bmul(invfact[i - 1], invs[i])
end

local function getPerm(n, k)
  if n == 0 then return 1 end
  if k == n then return fact[n] end
  return bmul(fact[n], invfact[n - k])
end

local function comp(a, b)
  return a < b
end

local function lower_bound(ary, x)
  local num = #ary
  if num == 0 then return 1 end
  if not comp(ary[1], x) then return 1 end
  if comp(ary[num], x) then return num + 1 end
  local min, max = 1, num
  while 1 < max - min do
    local mid = mfl((min + max) / 2)
    if comp(ary[mid], x) then
      min = mid
    else
      max = mid
    end
  end
  return max
end

local function upper_bound(ary, x)
  local num = #ary
  if num == 0 then return 1 end
  if comp(x, ary[1]) then return 1 end
  if not comp(x, ary[num]) then return num + 1 end
  local min, max = 1, num
  while 1 < max - min do
    local mid = mfl((min + max) / 2)
    if not comp(x, ary[mid]) then
      min = mid
    else
      max = mid
    end
  end
  return max
end

local n, x = io.read("*n", "*n")
local s = {}
local tbl = {}
for i = 1, n do
  s[i] = io.read("*n")
  tbl[i] = {}
end
table.sort(s)
for i = n, 1, -1 do
  local rem = x % s[i]
  if 0 < rem then
    tbl[i][rem] = getPerm(n - 1, n - i)
  end
end
local ret = 0
for src = n, 2, -1 do
  for val, cnt in pairs(tbl[src]) do
    local ub = upper_bound(s, val)
    if ub == 1 then
      ret = badd(ret, bmul(val, bmul(fact[src - 1], cnt)))
    else
      local skipfact = getPerm(src - 1, src - ub)
      cnt = bmul(cnt, skipfact)
      for dst = ub - 1, 1, -1 do
        local vc = bmul(cnt, getPerm(ub - 2, ub - 1 - dst))
        local rem = val % s[dst]
        if 0 < rem then
          if tbl[dst][rem] then
            tbl[dst][rem] = badd(tbl[dst][rem], vc)
          else
            tbl[dst][rem] = vc
          end
        end
      end
    end
  end
end
for val, cnt in pairs(tbl[1]) do
  ret = badd(ret, bmul(val, cnt))
end
print(ret)