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 Haskell (GHC 7.10.3) (judge verdict: Accepted)

import Data.Int
import Data.List
import Data.Vector.Unboxed ((!))
import qualified Data.Vector.Unboxed as V

main = do
  [n,x] <- map read.words <$> getLine
  set <- sortBy (flip compare).map read.words <$> getLine
  print $ solve n x set

solve :: Int64 -> Int64 -> [Int64] -> Int64
solve n x = (*% fact n).(!fromIntegral x).go 1
  where
  go _ [] = V.fromListN 100001 [0..100000]
  go i (s:set) = V.generate 100001 f
    where
    f x' = let x = fromIntegral x' in (a!fromIntegral x *% (n-i) + a!fromIntegral (x`mod`s)) *% inv (n-i+1)
    a = go (i+1) set

infixl 7 *%
infixl 5 +%
infixl 5 -%
a *% b = a * b `mod` 1000000007
a -% b = (a - b) `mod` 1000000007
a +% b = (a + b) `mod` 1000000007
fact = foldl (*%) 1.enumFromTo 2
inv x = u where (u,_,_) = euc x 1000000007
euc = go 1 0 0 1
  where
  go u v u' v' x 0 = (u,v,x)
  go u v u' v' x y = go u' v' (u - q * u') (v - q * v') y r
    where (q,r) = quotRem x y