P10782 [MX-J1-T2] "FLA - III" Ilumina

Original problem link: .

Define $\lfloor X \rfloor$ as the greatest integer less than or equal to $X$. For example, $\lfloor 1.99 \rfloor = 1, \lfloor 7 \rfloor = 7$. There are five positive integers $n, m, a, b, c$, and it is known that $b = \left\lfloor \frac{a}{n} \right\rfloor$, $c = \left\lfloor \frac{b}{m} \right\rfloor$. Given the values of $a$ and $c$, find a valid value of $b$, or report that no valid value of $b$ exists. **This problem uses a custom checker. If multiple valid values of $b$ exist, output any one of them.**

**This problem contains multiple test cases.** The first line contains a positive integer $T$, the number of test cases. For each test case, input one line with two positive integers $a, c$.

For each test case, output one integer on a single line. If no valid value of $b$ exists, output `-1`; otherwise output a valid value of $b$. **This problem uses a custom checker. If multiple valid values of $b$ exist, output any one of them.**

**"Sample Explanation #1"** For the first test case, choosing $n = 1, m = 1$ yields $b = 1$. For the second test case, choosing $n = 2, m = 1$ yields $b = 3$. For the third and fourth test cases, it can be proven that no valid value of $b$ exists. **Constraints** |Test Point ID|$a, c \leq$|Special Property| |:-:|:-:|:-:| |$1 \sim 3$|$10$|None| |$4 \sim 6$|$10^3$|None| |$7 \sim 8$|$10^{18}$|Yes| |$9 \sim 10$|$10^{18}$|None| Special property: It is guaranteed that for each given $a, c$, a valid value of $b$ always exists. For $100\%$ of the testdata, $1 \leq T \leq 10^5$, $1 \leq a, c \leq 10^{18}$. July 15, 2024: Added 7 hack testdata cases to Subtask #1. Translated by ChatGPT 5