310695: CF1872D. Plus Minus Permutation

D. Plus Minus Permutationtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard output

You are given $3$ integers — $n$, $x$, $y$. Let's call the score of a permutation$^\dagger$ $p_1, \ldots, p_n$ the following value:

$$(p_{1 \cdot x} + p_{2 \cdot x} + \ldots + p_{\lfloor \frac{n}{x} \rfloor \cdot x}) - (p_{1 \cdot y} + p_{2 \cdot y} + \ldots + p_{\lfloor \frac{n}{y} \rfloor \cdot y})$$

In other words, the score of a permutation is the sum of $p_i$ for all indices $i$ divisible by $x$, minus the sum of $p_i$ for all indices $i$ divisible by $y$.

You need to find the maximum possible score among all permutations of length $n$.

For example, if $n = 7$, $x = 2$, $y = 3$, the maximum score is achieved by the permutation $[2,\color{red}{\underline{\color{black}{6}}},\color{blue}{\underline{\color{black}{1}}},\color{red}{\underline{\color{black}{7}}},5,\color{blue}{\underline{\color{red}{\underline{\color{black}{4}}}}},3]$ and is equal to $(6 + 7 + 4) - (1 + 4) = 17 - 5 = 12$.

$^\dagger$ A permutation of length $n$ is an array consisting of $n$ distinct integers from $1$ to $n$ in any order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation (the number $2$ appears twice in the array) and $[1,3,4]$ is also not a permutation ($n=3$, but the array contains $4$).

Input

The first line of input contains an integer $t$ ($1 \le t \le 10^4$) — the number of test cases.

Then follows the description of each test case.

The only line of each test case description contains $3$ integers $n$, $x$, $y$ ($1 \le n \le 10^9$, $1 \le x, y \le n$).

Output

For each test case, output a single integer — the maximum score among all permutations of length $n$.

ExampleInput

8
7 2 3
12 6 3
9 1 9
2 2 2
100 20 50
24 4 6
1000000000 5575 25450
4 4 1

Output

12
-3
44
0
393
87
179179179436104
-6

Note

The first test case is explained in the problem statement above.

In the second test case, one of the optimal permutations will be $[12,11,\color{blue}{\underline{\color{black}{2}}},4,8,\color{blue}{\underline{\color{red}{\underline{\color{black}{9}}}}},10,6,\color{blue}{\underline{\color{black}{1}}},5,3,\color{blue}{\underline{\color{red}{\underline{\color{black}{7}}}}}]$. The score of this permutation is $(9 + 7) - (2 + 9 + 1 + 7) = -3$. It can be shown that a score greater than $-3$ can not be achieved. Note that the answer to the problem can be negative.

In the third test case, the score of the permutation will be $(p_1 + p_2 + \ldots + p_9) - p_9$. One of the optimal permutations for this case is $[9, 8, 7, 6, 5, 4, 3, 2, 1]$, and its score is $44$. It can be shown that a score greater than $44$ can not be achieved.

In the fourth test case, $x = y$, so the score of any permutation will be $0$.

题目大意：给定三个整数 n, x, y。排列 p_1, …, p_n 的分数定义为：

$$
\left( p_{1 \cdot x} + p_{2 \cdot x} + \ldots + p_{\lfloor \frac{n}{x} \rfloor \cdot x} \right) - \left( p_{1 \cdot y} + p_{2 \cdot y} + \ldots + p_{\lfloor \frac{n}{y} \rfloor \cdot y} \right)
$$

即排列的分数是所有索引 i 能被 x 整除的 p_i 之和，减去所有索引 i 能被 y 整除的 p_i 之和。

你需要找到长度为 n 的所有排列中可能的最大分数。

输入输出数据格式：

输入：
- 第一行包含一个整数 t (1 ≤ t ≤ 10^4) —— 测试用例的数量。
- 然后是每个测试用例的描述。
- 每个测试用例的描述仅包含一行，包含 3 个整数 n, x, y (1 ≤ n ≤ 10^9, 1 ≤ x, y ≤ n)。

输出：
- 对于每个测试用例，输出一个整数 —— 长度为 n 的所有排列中的最大分数。题目大意：给定三个整数 n, x, y。排列 p_1, …, p_n 的分数定义为： $$ \left( p_{1 \cdot x} + p_{2 \cdot x} + \ldots + p_{\lfloor \frac{n}{x} \rfloor \cdot x} \right) - \left( p_{1 \cdot y} + p_{2 \cdot y} + \ldots + p_{\lfloor \frac{n}{y} \rfloor \cdot y} \right) $$ 即排列的分数是所有索引 i 能被 x 整除的 p_i 之和，减去所有索引 i 能被 y 整除的 p_i 之和。 你需要找到长度为 n 的所有排列中可能的最大分数。 输入输出数据格式： 输入： - 第一行包含一个整数 t (1 ≤ t ≤ 10^4) —— 测试用例的数量。 - 然后是每个测试用例的描述。 - 每个测试用例的描述仅包含一行，包含 3 个整数 n, x, y (1 ≤ n ≤ 10^9, 1 ≤ x, y ≤ n)。 输出： - 对于每个测试用例，输出一个整数 —— 长度为 n 的所有排列中的最大分数。