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 (1.15.1) (judge verdict: Accepted)

use std::io::*;
use std::str::FromStr;

#[allow(dead_code)]
fn read<T: FromStr>() -> T {
    let stdin = stdin();
    let stdin = stdin.lock();
    let token: String = stdin
        .bytes()
        .map(|c| c.expect("failed to read char") as char)
        .skip_while(|c| c.is_whitespace())
        .take_while(|c| !c.is_whitespace())
        .collect();
    token.parse().ok().expect("failed to parse token")
}

fn main() {
    let n = read::<usize>();
    let k = read::<i64>();
    let mut seq = vec![0; n];
    let mut sum = 0;
    for i in 0..n {
        seq[i] = read::<i64>();
        sum += seq[i];
    }
    let mut candidates = vec![];
    for i in 1..sum {
        if i * i > sum {
            break;
        }
        if sum % i == 0 {
            candidates.push(i);
            candidates.push(sum / i);
        }
    }
    candidates.sort();
    candidates.reverse();
    for &m in candidates.iter() {
        let mut values = seq.iter().cloned().map(|x| x % m).collect::<Vec<_>>();
        values.sort();
        let mut l = 0;
        let mut r = n - 1;
        let mut lcnt = values[l];
        let mut rcnt = (m - values[r]) % m;
        while r - l > 1 {
            if lcnt <= rcnt {
                l += 1;
                lcnt += values[l];
            } else {
                r -= 1;
                rcnt += (m - values[r]) % m;
            }
        }
        // println!("{} {} {} {}", l, r, lcnt, rcnt);
        assert!(lcnt == rcnt);
        if lcnt <= k {
            println!("{}", m);
            return;
        }
    }
}