Max GCD

AtCoder

IDabc136_e

Time2000ms

Memory256MB

Difficulty—

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: $N$ $K$ $A_1$ $A_2$ $\cdots$ $A_{N-1}$ $A_{N}$ [samples]

Samples

Input #1

2 3
8 20

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 $A$.

Input #2

2 10
3 5

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 $A$.

Input #3

4 5
10 1 2 22

Output #3

Input #4

8 7
1 7 5 6 8 2 6 5

Output #4

API Response (JSON)

{
  "problem": {
    "name": "Max GCD",
    "description": {
      "content": "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 $i \\neq j$, e",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc136_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a sequence of $N$ integers: $A_1, A_2, \\cdots, A_N$.\nYou can perform the following operation between $0$ and $K$ times (inclusive):\n\n*   Choose two integers $i$ and $j$ such that $i \\neq j$, e...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments