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 Rust (1.42.0) (judge verdict: Accepted)

#![allow(unused_imports)]
use {
   proconio::{fastout, input, marker::*},
   std::cmp::*,
   std::collections::*,
};

#[fastout]
fn main() {
   input! {
      n:usize,
      k:usize,
      mut a:[usize; n]
   }
   let s: usize = a.iter().sum();
   let mut f = vec![];
   a.sort();
   for p in 1.. {
      if p * p >= s {
         if p * p == s {
            f.push(p);
         }
         break;
      }

      if s % p == 0 {
         f.push(p);
         f.push(s / p);
      }
   }

   f.sort_by(|a, b| b.cmp(a));
   for &d in f.iter() {
      let mut r = a.iter().map(|&x| x % d).collect::<Vec<_>>();
      r.sort();
      let mut x = vec![0; n + 1];
      let mut y = vec![0; n + 1];
      for i in 0..n {
         x[i + 1] = x[i] + r[i];
         y[i + 1] = y[i] + d - r[i];
      }
      let mut res = x[n];
      for i in 0..=n {
         res = min(res, max(x[i], y[n] - y[i]));
      }
      if res <= k {
         println!("{}", d);
         return;
      }
   }
}