Score : $600$ points

Problem Statement

There are $N$ integers $A_1, A_2, A_3, \dots, A_N$ written on a blackboard. You will do the following operation $N - 1$ times:

• Choose two numbers written on the blackboard and erase them. Let $x$ and $y$ be the erased numbers. Then, write $\gcd(x, y)$ or $\min(x, y)$ on the blackboard.

After $N - 1$ operations, just one integer will remain on the blackboard. How many possible values of this number are there?

Constraints

• $2 \le N \le 2000$
• $1 \le A_i \le 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$A_1$ $A_2$ $A_3$ $\dots$ $A_N$

Output

Print the number of possible values of the last remaining number on the blackboard.

3
6 9 12

2

The possible values of the last remaining number are $3$ and $6$. We will have $3$ in the end if, for example, we do as follows:

• choose $9, 12$ and erase them from the blackboard, then write $\gcd(9, 12) = 3$;
• choose $6, 3$ and erase them from the blackboard, then write $\min(6, 3) = 3$.

Also, we will have $6$ in the end if, for example, we do as follows:

• choose $6, 12$ and erase them from the blackboard, then write $\gcd(6, 12) = 6$;
• choose $6, 9$ and erase them from the blackboard, then write $\min(6, 9) = 6$.

4
8 2 12 6

1

$2$ is the only number that can remain on the blackboard.

7
30 28 33 49 27 37 48

7

$1$, $2$, $3$, $4$, $6$, $7$, and $27$ can remain on the blackboard.