5. Find all possible triples of positive integers, $a, b, c$ so that $\frac{a+1}{b}, \frac{b+1}{c}$, and $\frac{c+1}{a}$ are also integers.
so $a$ in $\{c-2, c-1, c, c+1\}$. Now a case bash yields 10 solutions:
$$
(1,2,3),(3,1,2),(2,3,1),(3,4,5),(5,3,4),(4,5,3),(2,1,1),(1,2,1),(1,1,2),(1,1,1) \text {. }
$$