Rectangle GCD

AtCoder

IDabc254_f

Time2000ms

Memory256MB

Difficulty—

You are given a positive integer $N$ and sequences of $N$ positive integers each: $A=(A_1,A_2,\dots,A_N)$ and $B=(B_1,B_2,\dots,B_N)$. We have an $N \times N$ grid. The square at the $i$\-th row from the top and the $j$\-th column from the left is called the square $(i,j)$. For each pair of integers $(i,j)$ such that $1 \le i,j \le N$, the square $(i,j)$ has the integer $A_i + B_j$ written on it. Process $Q$ queries of the following form. * You are given a quadruple of integers $h_1,h_2,w_1,w_2$ such that $1 \le h_1 \le h_2 \le N,1 \le w_1 \le w_2 \le N$. Find the greatest common divisor of the integers contained in the rectangle region whose top-left and bottom-right corners are $(h_1,w_1)$ and $(h_2,w_2)$, respectively. ## Constraints * $1 \le N,Q \le 2 \times 10^5$ * $1 \le A_i,B_i \le 10^9$ * $1 \le h_1 \le h_2 \le N$ * $1 \le w_1 \le w_2 \le N$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $Q$ $A_1$ $A_2$ $\dots$ $A_N$ $B_1$ $B_2$ $\dots$ $B_N$ $\mathrm{query}_1$ $\mathrm{query}_2$ $\vdots$ $\mathrm{query}_Q$ Each query is in the following format: $h_1$ $h_2$ $w_1$ $w_2$ [samples]

Samples

Input #1

Output #1

2
1
11
6
10

Let $C_{i,j}$ denote the integer on the square $(i,j)$.
For the $1$\-st query, we have $C_{1,2}=4,C_{1,3}=6,C_{2,2}=6,C_{2,3}=8$, so the answer is their greatest common divisor, which is $2$.

Input #2

Output #2

API Response (JSON)

{
  "problem": {
    "name": "Rectangle GCD",
    "description": {
      "content": "You are given a positive integer $N$ and sequences of $N$ positive integers each: $A=(A_1,A_2,\\dots,A_N)$ and $B=(B_1,B_2,\\dots,B_N)$. We have an $N \\times N$ grid. The square at the $i$\\-th row from ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc254_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a positive integer $N$ and sequences of $N$ positive integers each: $A=(A_1,A_2,\\dots,A_N)$ and $B=(B_1,B_2,\\dots,B_N)$.\nWe have an $N \\times N$ grid. The square at the $i$\\-th row from ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments