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)

use std::io;
use std::io::BufRead;

fn main() {
    let stdin = io::stdin();
    let mut stdin = stdin.lock();
    
    let (n, k) = {
        let mut input = String::new();
        stdin.read_line(&mut input).unwrap();
        let mut input = input.trim().split_whitespace();
        (
            input.next().unwrap().parse::<i32>().unwrap(),
            input.next().unwrap().parse::<i32>().unwrap(),
        )
    };

    let a: Vec<i32> = {
        let mut input = String::new();
        stdin.read_line(&mut input).unwrap();
        input.trim().split_whitespace().map(|s| s.parse::<i32>().unwrap()).collect()
    };

    let total: i32 = a.iter().sum();
    
    for p in (1..total+1).rev() {
        // let distance: i32 = a.iter().map(|i| {
        //     let n = i/p;
        //     let a = i-n*p;
        //     let b = (n+1)*p-i;
        //     if a > b { b } else { -a }
        // }).sum();
        
        // if distance%p == 0 {
            let distance: i32 = a.iter().map(|i| {
                let n = i/p;
                let a = i-n*p;
                let b = (n+1)*p-i;
                if a > b { b } else { a }
            }).sum();

            if distance <= k {
                if distance %p == 0 {
                    println!("{}", p);
                    return;
                }
            }
        // }
    }
}


fn distance( i: &i32, p: &i32 ) -> i32 {
    let n = i/p;
    let a = i-n*p;
    let b = (n+1)*p-i;
    if a > b { b } else { -a }
}