Score : $500$ points

问题描述

你将获得一个正整数 $N$ 以及两个各有 $N$ 个正整数的序列：$A=(A_1,A_2,\dots,A_N)$ 和 $B=(B_1,B_2,\dots,B_N)$。

我们有一个 $N \times N$ 的网格。从上到下第 $i$ 行、从左到右第 $j$ 列的方格被称为方格 $(i,j)$。对于每一对整数 $(i,j)$，满足 $1 \le i,j \le N$，方格 $(i,j)$ 上写有整数 $A_i + B_j$。处理 $Q$ 个以下形式的查询请求。

• 给定一组四个整数 $h_1,h_2,w_1,w_2$，满足 $1 \le h_1 \le h_2 \le N,1 \le w_1 \le w_2 \le N$。找出以 $(h_1,w_1)$ 为左上角、$(h_2,w_2)$ 为右下角的矩形区域内的所有整数的最大公约数。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

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:

$h_1$ $h_2$ $w_1$ $w_2$

Output

Print $Q$ lines. The $i$-th line should contain the answer to $\mathrm{query}_i$.

Sample Input 1

3 5
3 5 2
8 1 3
1 2 2 3
1 3 1 3
1 1 1 1
2 2 2 2
3 3 1 1

Sample 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 $2$.

Sample Input 2

1 1
9
100
1 1 1 1

Sample Output 2

109

update @ 2024/3/10 10:50:11