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F - Rectangle GCD

Score : $500$ points

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.

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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 $\mathrm{query}_i$.

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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

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 $2$.

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1 1
9
100
1 1 1 1

109