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)]
#![allow(non_snake_case)]
use std::cmp::*;
use std::collections::*;
use std::io::Write;

#[allow(unused_macros)]
macro_rules! debug {
    ($($e:expr),*) => {
        #[cfg(debug_assertions)]
        $({
            let (e, mut err) = (stringify!($e), std::io::stderr());
            writeln!(err, "{} = {:?}", e, $e).unwrap()
        })*
    };
}

fn main() {
    let v = read_vec::<i64>();
    let (n, k) = (v[0] as usize, v[1]);
    let a = read_vec::<i64>();
    let a_sum = a.iter().sum::<i64>();
    let mut cands = vec![];
    for i in 1.. {
        if i * i > a_sum {
            break;
        }
        if a_sum % i == 0 {
            cands.push(i);
            cands.push(a_sum / i);
        }
    }
    cands.sort();
    cands.dedup();
    cands.reverse();

    for cand in cands {
        if ok(&a, cand, k) {
            println!("{}", cand);
            return;
        }
    }
}

fn ok(a: &Vec<i64>, cand: i64, k: i64) -> bool {
    let mut a_clone = a
        .iter()
        .cloned()
        .map(|x| x % cand)
        .filter(|&x| x != 0)
        .collect::<Vec<_>>();
    a_clone.sort();

    if a_clone.is_empty() {
        return true;
    }
    // debug!(a_clone);
    let n = a_clone.len();
    let mut minus_from_left = vec![0; n];
    minus_from_left[n - 1] = cand - a_clone[n - 1];
    for i in (0..n - 1).rev() {
        minus_from_left[i] = minus_from_left[i + 1] + (cand - a_clone[i]);
    }
    let mut accum = 0;
    for i in 0..n - 1 {
        accum += a_clone[i];
        if accum == minus_from_left[i + 1] {
            return accum <= k;
        }
    }
    unreachable!();
}

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}