Score : $400$ points

问题描述

有 $N$ 个正方形按顺序排列，编号从 $0$ 到 $(N-1)$。Snuke 将按照以下步骤标记每个正方形。

1. 标记编号为 $0$ 的正方形。
2. 重复执行以下步骤 i - iii 共 $(N-1)$ 次。
   1. 将变量 $x$ 初始化为 $(A+D) \bmod N$，其中 $A$ 是上次标记的正方形的索引。
   2. 当正方形 $x$ 已被标记时，重复将 $x$ 替换为 $(x+1) \bmod N$。
   3. 标记正方形 $x$。

找出 Snuke 第 $K$ 次标记的正方形的索引。

给定 $T$ 个测试用例，请分别找出它们的答案。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There are $N$ squares indexed $0$ through $(N-1)$ arranged in a line. Snuke is going to mark every square by the following procedure.

1. Mark square $0$.
2. Repeat the following steps i - iii $(N-1)$ times.
   1. Initialize a variable $x$ with $(A+D) \bmod N$, where $A$ is the index of the square marked last time.
   2. While square $x$ is marked, repeat replacing $x$ with $(x+1) \bmod N$.
   3. Mark square $x$.

Find the index of the square that Snuke marks for the $K$-th time.

Given $T$ test cases, find the answer for each of them.

Constraints

• $1\leq T \leq 10^5$
• $1\leq K\leq N \leq 10^9$
• $1\leq D \leq 10^9$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format, where $\mathrm{test}_i$ denotes the $i$-th test case:

$T$

$\mathrm{test}_1$

$\mathrm{test}_2$

$\vdots$

$\mathrm{test}_T$

Each test case is given in the following format:

$N$ $D$ $K$

Output

Print $T$ lines.

The $i$-th $(1\leq i \leq T)$ line should contain the answer to the $i$-th test case.

Sample Input 1

9
4 2 1
4 2 2
4 2 3
4 2 4
5 8 1
5 8 2
5 8 3
5 8 4
5 8 5

Sample Output 1

0
2
1
3
0
3
1
4
2

If $N=4$ and $D=2$, Snuke marks the squares as follows.

1. Mark square $0$.
2. ($1$-st iteration) Let $x=(0+2)\bmod 4=2$. Since square $2$ is not marked, mark it.
   ($2$-nd iteration) Let $x=(2+2)\bmod 4=0$. Since square $0$ is marked, let $x=(0+1)\bmod 4=1$. Since square $1$ is not marked, mark it.
   ($3$-rd iteration) Let $x=(1+2)\bmod 4=3$. Since square $3$ is not marked, mark it.

update @ 2024/3/10 12:07:04