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: Wrong Answer)

fn read_stdin() -> String {
    use std::io::Read;
    let mut input = String::new();
    std::io::stdin().read_to_string(&mut input).unwrap();
    input
}

fn scan<'a, I: Iterator<Item = &'a str>, T: std::str::FromStr>(tokens: &mut I) -> T {
    tokens.next().unwrap().parse().ok().unwrap()
}

fn vec<T, F: FnMut(usize) -> T>(n: usize, mut f: F) -> Vec<T> {
    (0..n).map(|i| f(i)).collect()
}

fn main() {
    let input = read_stdin();
    let tokens = &mut input.split_whitespace();
    let n: usize = scan(tokens);
    let k: usize = scan(tokens);
    let a: Vec<usize> = vec(n, |_| scan(tokens));
    let sum: usize = a.iter().sum();
    let max: usize = *a.iter().max().unwrap();
    if sum - max <= k {
        println!("{}", sum);
        return;
    }
    let mut l: usize = 1;
    let mut u: usize = sum;
    let mut r: Vec<usize> = vec![0; n];
    while l + 1 < u {
        let c = (l + u) / 2;
        let mut x = c;
        let mut y = sum / x;
        while x * y < sum {
            x = (sum + y - 1) / y;
            y = sum / x;
        }
        if x >= u {
            u = c;
            continue;
        }
        for i in 0..n {
            r[i] = a[i] % x;
        }
        r.sort();
        let mut i = 0;
        let mut j = n - 1;
        let mut cnt = 0;
        while i < j {
            r[j] += r[i];
            cnt += r[i];
            i += 1;
            while r[j] >= x {
                r[j - 1] += r[j] % x;
                j -= 1;
            }
        }
        if cnt <= k {
            l = x;
        } else {
            u = c;
        }
    }
    println!("{}", l);
}