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)

import java.io.*
import java.util.*

val pw = PrintWriter(System.out)
const val MOD = 1000000007L
const val INF = 2147483647
const val LINF = 9223372036854775807L

fun main(args: Array<String>) {
    solve()
    pw.flush()
}

fun solve() {
    val tmp = nextIntList()
    val k = tmp[1]
    val a = nextIntList()
    val s = a.sum()
    val div: MutableList<Int> = mutableListOf()
    for (i in 1..s) {
        if(i * i > s) break
        if(s % i == 0){
            div.add(i)
            div.add(s / i)
        }
    }

    var ans = 1
    for (d in div){
        val sp: MutableList<Int> = mutableListOf()
        for (i in a){
            sp.add(i % d)
        }
        sp.sortDescending()
        if(sp.sum() / 2  <= k){
            ans = max(ans, d)
        }
    }

    println(ans)

}


fun println(v: String) {
    pw.println(v)
}

fun print(v: String) {
    pw.print(v)
}

fun next() = readLine()!!
fun nextInt() = next().toInt()
fun nextLong() = next().toLong()
fun nextDouble() = next().toDouble()
fun nextList() = next().split(" ").toMutableList()
fun nextIntList(index: Int = 0) = next().split(" ").map { it.toInt() + index }.toMutableList()
fun nextLongList(index: Long = 0L) = next().split(" ").map { it.toLong() + index }.toMutableList()
fun nextDoubleList() = next().split(" ").map { it.toDouble() }.toMutableList()

fun ary(n: Int, init: String = "") = Array(n) { init }
fun intAry(n: Int, init: Int = 0) = IntArray(n) { init }
fun longAry(n: Int, init: Long = 0L) = LongArray(n) { init }
fun doubleAry(n: Int, init: Double = 0.0) = DoubleArray(n) { init }
fun ary2(n: Int, m: Int, init: String = "") = Array(n) { Array(m) { init } }
fun intAry2(n: Int, m: Int, init: Int = 0) = Array(n) { IntArray(m) { init } }
fun longAry2(n: Int, m: Int, init: Long = 0) = Array(n) { LongArray(m) { init } }
fun doubleAry2(n: Int, m: Int, init: Double = 0.0) = Array(n) { DoubleArray(m) { init } }
fun ary3(n: Int, m: Int, k: Int, init: String = "") = Array(n) { Array(m) { Array(k) { init } } }
fun intAry3(n: Int, m: Int, k: Int, init: Int = 0) = Array(n) { Array(m) { IntArray(k) { init } } }
fun longAry3(n: Int, m: Int, k: Int, init: Long = 0L) = Array(n) { Array(m) { LongArray(k) { init } } }
fun doubleAry3(n: Int, m: Int, k: Int, init: Double = 0.0) = Array(n) { Array(m) { DoubleArray(k) { init } } }

fun abs(n: Int): Int = Math.abs(n)
fun abs(n: Long): Long = Math.abs(n)
fun abs(n: Double): Double = Math.abs(n)
fun max(a: Int = -INF, b: Int = -INF, c: Int = -INF, d: Int = -INF, e: Int = -INF): Int = listOf(a, b, c, d, e).max()!!
fun max(a: Long = -LINF, b: Long = -LINF, c: Long = -LINF, d: Long = -LINF, e: Long = -LINF): Long =
    listOf(a, b, c, d, e).max()!!.toLong()

fun min(a: Int = INF, b: Int = INF, c: Int = INF, d: Int = INF, e: Int = INF): Int = listOf(a, b, c, d, e).min()!!
fun min(a: Long = LINF, b: Long = LINF, c: Long = LINF, d: Long = LINF, e: Long = LINF): Long =
    listOf(a, b, c, d, e).min()!!.toLong()