P6810 "MCOI-02" Convex Hull

A contest needs an easy check-in problem.

Leasier was caught playing Minecraft, so he had to compute the value of the following expression. $$\displaystyle\sum_{i = 1}^n \sum_{j = 1}^m \tau(i) \tau(j) \tau(\gcd(i, j))$$ Since the result may be very large, you only need to output the value modulo $p$. If you have questions about the mathematical symbols in this problem, please check the "Hint" section.

One line with three integers $n, m, p$.

One line with one integer, representing the required value.

#### Constraints and Notes **This problem uses bundled testdata.** | Subtask | $n, m$ | Score | | :------: | :------: | :------: | | $1$ | $1 \leq n, m \leq 10^3$ | $15 \operatorname{pts}$ | | $2$ | $1 \leq n, m \leq 10^5$ | $25 \operatorname{pts}$ | | $3$ | $1 \leq n, m \leq 10^6$ | $30 \operatorname{pts}$ | | $4$ | No special constraints | $30 \operatorname{pts}$ | For $100\%$ of the testdata, $1 \leq n, m \leq 2 \times 10^6$, and $1 \leq p \leq 10^9$. #### Hint As a beginner-friendly check-in problem, of course hints are provided. - $\sum$ is the summation symbol. For example, $\displaystyle\sum_{i = 1}^n i$ means $1 + 2 + \cdots + n$. - $\tau$ denotes the number of divisors. For example, $\tau(6) = 4$. - $\gcd$ is the greatest common divisor. For example, $\gcd(12, 15) = 3$. #### Notes Minecraft OI Round 2 A - Maker: Leasier - Tester: happydef Translated by ChatGPT 5