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 Kotlin (1.0.0) (judge verdict: Runtime Error)

import java.math.*

fun main(args: Array<String>) {
    val (n, k) = readInputLine().split(" ").map{it.toInt()}
    val aArr = readInputLine().split(" ").map{it.toInt()}
    
    // dp[数][足した][引いた]
    val dp = Array(n, {Array(k, {IntArray(k)})})

    for (i in 0..k) {
        dp[0][i][0] = aArr[i] + i
    }
    
    for (i in 0..k) {
        dp[0][0][i] = Math.abs(aArr[0] - i)
    }
    
    for (i in 1 until n) {
        dp[i][0][0] = gcd(dp[i - 1][0][0], aArr[i])
        for (j in 1..k) {
            for (l in 0..k) {
                dp[i][j][l] = gcd(dp[i - 1][j - 1][l], aArr[i + 1])
            }
        }
        for (j in 0..k) {
            for (l in 1..k) {
                dp[i][j][l] = gcd(dp[i - 1][j][l - 1], Math.abs(aArr[i] - 1))
            }
        }
    }

    var ans = 0
    
    for (i in 0..k) {
        for (j in 0..k) {
            if (ans < dp[n - 1][i][j]) {
                ans = dp[n - 1][i][j]
            }
        }
    }

    println(ans)
}

fun readInputLine(): String {
    return readLine()!!
}

fun gcd(a: Int, b: Int): Int {
    if (b > a) return gcd(b, a)
    // a >= b とする
    if (b == 0) {
        return a
    }
    return gcd(b, a % b)
}