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: Wrong Answer)

fun main(args: Array<String>) {
    fun divisor(n: Long): List<Long> {
        val list = mutableListOf<Long>()
        var i = 1L
        while (i * i <= n) {
            if (n % i == 0L) {
                list.add(i)
                if (i != n / i) {
                    list.add(n / i)
                }
            }
            i++
        }
        return list.sortedDescending()
    }

    val (n, k) = readLine()!!.split(' ').map(String::toLong)
    val aList = readLine()!!.split(' ').map(String::toLong)
    val divList = divisor(aList.sum())

    for (div in divList) {
        var moveCountM = 0L
        var moveCountP = 0L
        for (i in 0 until n.toInt()) {
            val a = aList[i]
            val move = a % div
            if (div / 2 > move) {
                moveCountM += move
                if (moveCountM > k) {
                    moveCountM -= move
                    moveCountP += div - move
                }
            } else {
                moveCountP += div - move
                if (moveCountP > k) {
                    moveCountP -= div - move
                    moveCountM += move
                }
            }
            if (moveCountM > k || moveCountP > k) {
                break
            }
        }
        if (moveCountM > k || moveCountP > k) {
            continue
        } else {
            println(div)
            return
        }
    }
    println(1)
}