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 Go (1.14.1) (judge verdict: Accepted)

package main

import (
	"bufio"
	"fmt"
	"os"
	"sort"
	"strconv"
)

var sc = bufio.NewScanner(os.Stdin)

func Scan() string {
	sc.Scan()
	return sc.Text()
}
func iScan() int {
	n, _ := strconv.Atoi(Scan())
	return n
}
func fScan() float64 {
	n, _ := strconv.ParseFloat(Scan(), 64)
	return n
}
func stringToInt(s string) int {
	n, _ := strconv.Atoi(s)
	return n
}
func SScan(n int) []string {
	a := make([]string, n)
	for i := 0; i < n; i++ {
		a[i] = Scan()
	}
	return a
}
func iSScan(n int) []int {
	a := make([]int, n)
	for i := 0; i < n; i++ {
		a[i] = iScan()
	}
	return a
}
func abs(x int) int {
	if x < 0 {
		x = -x
	}
	return x
}
func larger(a, b int) int {
	if a < b {
		return b
	} else {
		return a
	}
}
func smaller(a, b int) int {
	if a > b {
		return b
	} else {
		return a
	}
}
func max(a []int) int {
	x := a[0]
	for i := 0; i < len(a); i++ {
		if x < a[i] {
			x = a[i]
		}
	}
	return x
}
func min(a []int) int {
	x := a[0]
	for i := 0; i < len(a); i++ {
		if x > a[i] {
			x = a[i]
		}
	}
	return x
}
func sum(a []int) int {
	x := 0
	for _, v := range a {
		x += v
	}
	return x
}
func fSum(a []float64) float64 {
	x := 0.
	for _, v := range a {
		x += v
	}
	return x
}

var mod int = 1000000007

func main() {
	buf := make([]byte, 0)
	sc.Buffer(buf, mod)
	sc.Split(bufio.ScanWords)
	n, k := iScan(), iScan()
	a := iSScan(n)
	x := sum(a)
	d := make([]int, 0)
	for i := 1; i*i <= x; i++ {
		if x%i == 0 {
			d = append(d, i)
			if i != x/i {
				d = append(d, x/i)
			}
		}
	}
	sort.Sort(sort.Reverse(sort.IntSlice(d)))
	for _, v := range d {
		y := make([]int, n)
		z := make([]int, n)
		for i := 0; i < n; i++ {
			y[i] = a[i] % v
		}
		sort.Ints(y)
		yy := make([]int, n)
		copy(yy, y)
		for i := 0; i < n; i++ {
			z[i] = v - y[i]
		}
		for i := 1; i < n; i++ {
			y[i] += y[i-1]
		}
		for i := n - 2; i >= 0; i-- {
			z[i] += z[i+1]
		}
		c := 0
		for i := 0; i < n-1; i++ {
			c += yy[i]
			if y[i] == z[i+1] {
				break
			}
		}
		if c <= k {
			fmt.Println(v)
			break
		}
	}
}