201262: [AtCoder]ARC126 C - Maximize GCD

Problem Statement

Given is a sequence of $N$ positive integers: $A = (A_1, A_2, \ldots, A_N)$. You can do the following operation on this sequence at least zero and at most $K$ times:

• choose $i\in \{1,2,\ldots,N\}$ and add $1$ to $A_i$.

Find the maximum possible value of $\gcd(A_1, A_2, \ldots, A_N)$ after your operations.

Constraints

• $2\leq N\leq 3\times 10^5$
• $1\leq K\leq 10^{18}$
• $1 \leq A_i\leq 3\times 10^5$

Input

Input is given from Standard Input in the following format:

$N$ $K$
$A_1$ $A_2$ $\ldots$ $A_N$

Output

Print the maximum possible value of $\gcd(A_1, A_2, \ldots, A_N)$ after your operations.

Sample Input 1

3 6
3 4 9

Sample Output 1

5

One way to achieve $\gcd(A_1, A_2, A_3) = 5$ is as follows.

• Do the operation with $i = 1$ twice, with $i = 2$ once, and with $i = 3$ once, for a total of four times, which is not more than $K=6$.
• Now we have $A_1 = 5$, $A_2 = 5$, $A_3 = 10$, for which $\gcd(A_1, A_2, A_3) = 5$.

Sample Input 2

3 4
30 10 20

Sample Output 2

10

Doing no operation achieves $\gcd(A_1, A_2, A_3) = 10$.

Sample Input 3

5 12345
1 2 3 4 5

Sample Output 3

2472