Problem:
Score : 500 points

Problem Statement

We have a sequence of N integers: A_1, A_2, \cdots, A_N.

You can perform the following operation between 0 and K times (inclusive):

Choose two integers i and j such that i \neq j, each between 1 and N (inclusive). Add 1 to A_i and -1 to A_j, possibly producing a negative element.

Compute the maximum possible positive integer that divides every element of A after the operations. Here a positive integer x divides an integer y if and only if there exists an integer z such that y = xz.

Constraints

2 \leq N \leq 500

1 \leq A_i \leq 10^6

0 \leq K \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K
A_1 A_2 \cdots A_{N-1} A_{N}

Output

Print the maximum possible positive integer that divides every element of A after the operations.

Sample Input 1

2 3
8 20

Sample Output 1

7

7 will divide every element of A if, for example, we perform the following operation:

Choose i = 2, j = 1. A becomes (7, 21).

We cannot reach the situation where 8 or greater integer divides every element of A.

Sample Input 2

2 10
3 5

Sample Output 2

8

Consider performing the following five operations:

Choose i = 2, j = 1. A becomes (2, 6).

Choose i = 2, j = 1. A becomes (1, 7).

Choose i = 2, j = 1. A becomes (0, 8).

Choose i = 2, j = 1. A becomes (-1, 9).

Choose i = 1, j = 2. A becomes (0, 8).

Then, 0 = 8 \times 0 and 8 = 8 \times 1, so 8 divides every element of A. We cannot reach the situation where 9 or greater integer divides every element of A.

Sample Input 3

4 5
10 1 2 22

Sample Output 3

7

Sample Input 4

8 7
1 7 5 6 8 2 6 5

Sample Output 4

5

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <iostream>
#include <sstream>
#include <iterator>
#include <algorithm>
#include <vector>
#include <map>

using namespace std;
 
int main(){
  int N,K;
  cin>>N>>K;
  vector<int> A(N,0);
  for(int i=0;i<N;++i){
    cin>>A[i];
  }
  
  int total=0;
  for(int i=0;i<N;++i){
    total+=A[i];
  }
  
  for(int div=1;div<=total/2;++div){
    if(total%div!=0){
      continue;
    }
    vector<int> mod(N,0);
    int num=total/div;
    for(int i=0;i<N;++i){
      mod[i]=A[i]%num;
    }
    
    sort(mod.begin(),mod.end());
    int count=0;
    while(count<K){
      if(mod.size()==0){
        cout<<num<<endl;
        return 0;
      }else if(mod[0]==0){
        mod.erase(mod.begin());
        continue;
      }
      int step=min(mod[0],num-mod[mod.size()-1]);
      count+=step;
      mod[0]-=step;
      if(mod[0]==0){
        mod.erase(mod.begin());
      }
      mod[mod.size()-1]+=step;
      if(mod[mod.size()-1]==num){
        mod.erase(mod.end()-1);
      }
    }
  }
  cout<<"1"<<endl;
  return 0;
}