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: Runtime Error)

#[allow(dead_code)]
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()
}
#[allow(dead_code)]
fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}
#[allow(dead_code)]
fn read_vec2<T: std::str::FromStr>(n: u32) -> Vec<Vec<T>> {
    (0..n).map(|_| read_vec()).collect()
}
#[allow(dead_code)]
fn yn(result: bool) {
    if result {
        println!("Yes");
    } else {
        println!("No");
    }
}
fn gcd(x: u64, y: u64) -> u64 {
    if x == 0 {
        y
    } else {
        gcd(y % x, x)
    }
}
fn main() {
    let v = read_vec::<usize>();
    let n = v[0];
    let k = v[1];
    let a = read_vec::<u64>();

    let mut dp = vec![vec![0u64; k+1]; n+1];
    dp[0][0] = 0;
    for j in 0..k+1 {
        dp[1][j] = a[0] + j as u64;
    }
    for i in 1..n {
        for j in 0..k {
            dp[i + 1][j] = std::cmp::max(
                dp[i + 1][j], gcd(dp[i][j], a[i]));
            if j + a[i] as usize <= k {
                dp[i + 1][j + a[i] as usize] = std::cmp::max(
                    dp[i + 1][j + a[i] as usize], dp[i][j] + a[i]
                );
            }
        }
    }
    println!("{}", dp[n][k]);

    /* loop pattern
    let mut count = 0;
    loop {
        if  { break; }
        count += 1;
    }
    println!("{}", count);
    */

    /* yes/no pattern
    yn(a == b);
    */
}