MathNet 2024-part-04 — 168 problems (EN–VI) Question 168

Find all the numbers $n$ and $p$, with $n$ a positive integer and $p$ a prime, so that the numbers $\frac{p-2}{n}$, $\frac{n(n+2)}{p}$ and $\frac{n(n+2)+p-2}{n+p}$ are integers.