## Task 17/87

Determine all pairs $(p ; q)$ of prime numbers $p$ and $q$ that satisfy the following conditions:

1. Their sum $P$ is also a prime number.
2. The product of the three prime numbers $p ; q ; P$ is divisible by 10.
The following observations lead to the solution:

1. Due to the commutativity of addition and multiplication, if $\left(p_{1} ; q_{1}\right)$ is a solution, then $\left(p_{2}=q_{1} ; q_{2}=p_{1}\right)$ is also a solution.
2. Due to the uniqueness of prime factorization, among the primes $p ; q ; P=p+q$, the primes 2 and 5 must occur.
3. Clearly, $P>2$. Therefore, $p=2$ (or $q=2$).
4. The remaining possibilities are $q=5$ (or $p=5$), $P=7$, and $P=5, q=3$ (or $p=3$).

Thus, the solutions are $(2 ; 3),(3 ; 2),(2 ; 5),(5 ; 2)$. Further solutions are excluded by the solution process (completeness of the case distinction).