A16231. Plus Minus Permutation

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:

$$$$(p_{1 \cdot x} + p_{2 \cdot x} + \ldots + p_{\lfloor \frac{n}{x} \rfloor \cdot x}) - (p_{1 \cdot y} + p_{2 \cdot y} + \ldots + p_{\lfloor \frac{n}{y} \rfloor \cdot y}) $$ </span></p><p>In other words, the <span class="tex-font-style-it">score</span> 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 $y$ .</p><p>You need to find the maximum possible <span class="tex-font-style-it">score</span> among all permutations of length $n$ .</p><p>For example, if $n = 7$ , $x = 2$ , $y = 3$ , the maximum <span class="tex-font-style-it">score</span> 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 $(6 + 7 + 4) - (1 + 4) = 17 - 5 = 12$ .</p><p> $^\\dagger$ 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$$$).

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 $n$ , $x$ , $y$ ( $1 \le n \le 10^9$ , $1 \le x, y \le n$ ).

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

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 $0$ .