101364: [AtCoder]ABC136 E - Max GCD

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