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)

fn factor(mut x: usize) -> Vec<usize> {
    let mut div = 2;
    let mut results = vec![];
    while x > 0 {
        while x % div == 0 {
            results.push(div);
            x /= div;
        }
        div += 1;
        if div * div > x {
            break;
        }
    }
    if x != 1 {
        results.push(x);
    }
    results
}

fn ok(x: usize, k: usize, a: &[usize]) -> bool {
    let mut b = vec![];
    for &v in a.iter() {
        b.push(v % x);
    }
    b.sort();
    let n = b.len();
    let mut left = vec![0; n + 1];
    for i in 0..n {
        left[i + 1] = left[i] + b[i];
    }
    let mut right = vec![0; n + 1];
    for i in (0..n).rev() {
        right[i] = x - b[i] + right[i + 1];
    }

    // println!("{} {:?} {:?} {:?}", x, left, right, b);
    for i in 0..n + 1 {
        if std::cmp::max(left[i], right[i]) <= k {
            return true;
        }
    }

    false
}

fn solve() {
    let s = std::io::stdin();
    let mut sc = Scanner { stdin: s.lock() };
    let n: usize = sc.read();
    let k: usize = sc.read();
    let mut a: Vec<usize> = sc.vec(n);
    a.sort();
    let result = a.iter().fold(0, |sum, a| sum + a);
    let factors = factor(result);

    // println!("{:?}", factors);
    let mut result = 1;
    for bit in 0..(1usize << factors.len()) {
        let mut value = 1;
        for i in 0..factors.len() {
            if (bit >> i) & 1 != 0 {
                value *= factors[i];
            }
        }
        if ok(value, k, &a) {
            result = std::cmp::max(result, value);
        }
    }
    println!("{}", result);
}

fn main() {
    std::thread::Builder::new()
        .stack_size(64 * 1024 * 1024) // 64MB
        .spawn(|| solve())
        .unwrap()
        .join()
        .unwrap();
}

//snippet from kenkoooo
pub struct Scanner<R> {
    stdin: R,
}

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn read_line(&mut self) -> String {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b'\n' || b == b'\r')
            .take_while(|&b| b != b'\n' && b != b'\r')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }

    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}