Question 20
Let $p$ be an odd prime number and $a, b, c$ be integers so that the integers
$$
a^{2023} + b^{2023}, \quad b^{2024} + c^{2024}, \quad c^{2025} + a^{2025}
$$
are all divisible by $p$. Prove that $p$ divides each of $a, b$, and $c$.
$$
a^{2023} + b^{2023}, \quad b^{2024} + c^{2024}, \quad c^{2025} + a^{2025}
$$
are all divisible by $p$. Prove that $p$ divides each of $a, b$, and $c$.