GCD or MIN

AtCoder

IDabc191_f

Time2000ms

Memory256MB

Difficulty—

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: $N$ $A_1$ $A_2$ $A_3$ $\dots$ $A_N$ [samples]

Samples

Input #1

3
6 9 12

Output #1

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 $\min(6, 9) = 6$.

Input #2

4
8 2 12 6

Output #2

1

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

Input #3

7
30 28 33 49 27 37 48

Output #3

7

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

API Response (JSON)

{
  "problem": {
    "name": "GCD or MIN",
    "description": {
      "content": "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 $",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc191_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ integers $A_1, A_2, A_3, \\dots, A_N$ written on a blackboard.  \nYou will do the following operation $N - 1$ times:\n\n*   Choose two numbers written on the blackboard and erase them. Let $...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments