(R. Henner) Determine all triples $(a, b, c)$ of positive integers for which the conditions

$$
\begin{aligned}
\operatorname{gcd}(a, 20) & =b, \\
\operatorname{gcd}(b, 15) & =c \quad \text { and } \\
\operatorname{gcd}(a, c) & =5
\end{aligned}
$$

hold.
(R. Henner) From (III) it follows that $a$ is divisible by 5.

- Case $a$): Let $a$ be an odd number, so $a$ is a multiple of 5 but not of 10. Then it follows from (I): $b=5$, and from (II): $c=5$.
- Case b): Let $a$ be even but not divisible by 4, so $a$ is a multiple of 10 but not of 20. Then it follows from (I): $b=10$, and from (II): $c=5$.
- Case c): Let $a$ be divisible by 4, so a multiple of 20. Then it follows from (I): $b=20$, and from (II): $c=5$.

Thus, the solution set is $\{(20 t, 20,5),(20 t-10,10,5),(10 t-5,5,5) \mid t$ is a positive integer $\}$.