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 JavaScript (node.js v5.12) (judge verdict: Time Limit Exceeded)

    function main(input) {
        var data = input.split("\n").map(function(line){return line.split(" ").map(function(value){return parseInt(value)})})
        var k = data[0][1];
        var array = data[1];
        var maxA = max(array);
     
        var maxG = Math.min(maxA + k, sum(array))
        for (var i=maxG; i > 1; i--) {
        	if (isG(i, array, k)) {
        		return console.log(i);
        	}
        }
        return console.log(1);
    }
    function max(array) {
    	return array.reduce(function(a,b){ return Math.max(a, b)})
    }
    function sum(array) {
    	return array.reduce(function(current, value){return current + value}, 0)
    }
    function isG(g, array, k) {
    	var newArray = [];
    	var tempK = 2*k;
    	for (var i=0;i<array.length;i++) {
    		var mod1 = array[i] % g;
    		if (mod1==0) continue;
    		var mod2 = g - mod1;
    		if (k < Math.min(mod1, mod2)) return false;
    		tempK -= Math.min(mod1, mod2);
    		if (tempK < 0) return false;
    		newArray.push([mod1, mod2]);
    	}
    	return traver(newArray, 0, 0, 0, k)
    }
    function traver(array, level, leftCost, rightCost, budget) {
    	if (leftCost > budget) return false;
    	if (rightCost > budget) return false;
    	if (level == array.length) return leftCost == rightCost;
    	if (leftCost < rightCost) {
    		return testLeftPath(array, level, leftCost, rightCost, budget) || testRightPath(array, level, leftCost, rightCost, budget);
    	} else {
    		return testRightPath(array, level, leftCost, rightCost, budget) || testLeftPath(array, level, leftCost, rightCost, budget)
    	}
    	return false;
    }
    function testLeftPath(array, level, leftCost, rightCost, budget) {
    	return traver(array, level + 1, leftCost + array[level][0], rightCost, budget);
    }
    function testRightPath(array, level, leftCost, rightCost, budget) {
    	return traver(array, level + 1, leftCost, rightCost + array[level][1], budget);
    }
    main(require('fs').readFileSync('/dev/stdin', 'utf8'));