## Task A-2.3.

Let $p$ and $q$ be prime numbers such that $p+q+4$ and $p q-12$ are also prime numbers. Determine $p+q$.
## Solution.

Numbers $p$ and $q$ cannot have the same parity because then $p+q+4$ would be an even number strictly greater than 2. Therefore, one of the numbers $p$ or $q$ must be equal to 2.

Without loss of generality, we can assume that $p=2$.

According to the conditions of the problem, $q+6$ and $2q-12$ are prime numbers. The number $2q-12$ is clearly even, and since it is also prime, it must be $2q-12=2$, i.e., $q=7$.

The numbers $q=7$ and $p+q+4=13$ are also prime, so all conditions of the problem are satisfied. 1 point Therefore, the final value of $p+q=9$.

Note: The fifth point in the above scoring scheme can only be achieved if the solution verifies (or mentions) that all four numbers $p=2, q=7, p+q+4=13$, and $pq-12=2$ are indeed prime.