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)

use std::io;
use std::cmp;
use std::collections::BinaryHeap;
use std::collections::HashMap;
use std::collections::HashSet;

fn read_line() -> String {
    let mut buf = String::new();
    io::stdin().read_line(&mut buf).ok();
    buf
}

fn check(a: &Vec<i64>, tar: i64, k: i64) -> bool {
    let mut b = Vec::with_capacity(a.len());

    for &x in a.iter() {
        if x % tar != 0 {
            b.push(x % tar);
        }
    }
    b.sort();
    let mut c = vec![0; b.len()+1];
    let mut d = vec![0; b.len()+1];

    for i in 0..b.len() {
        c[i+1] = c[i] + b[i];
    }
    for i in (0..b.len()).rev() {
        d[i] = d[i+1] + (tar - b[i]);
    }
    for i in 0..b.len() {
        if (c[i] - d[i])%tar == 0 && cmp::max(c[i], d[i]) <= k {
            return true;
        }
    }
    false
}


fn main() {
    let mut line: Vec<i64> = read_line().split_whitespace().map(|s| s.parse().unwrap()).collect();
    let n = line[0] as usize;
    let k = line[1];
    let mut a: Vec<i64> = read_line().split_whitespace().map(|s| s.parse().unwrap()).collect();
    a.sort();

    let mut sum = 0;
    for i in a.iter() {
        sum += i;
    }

    let mut ans = 1;
    for i in 1..sum {
        if i*i > sum { 
            break;
        }
        if sum % i > 0 {
            continue;
        }
        let tar = sum / i;

        if check(&a, tar, k) {
            ans = tar;
            break;
        } else if check(&a, i, k) {
            ans = i;
        }
    }
    println!("{}", ans);
}