## Task 4 - 280734

Determine all pairs $(p, q)$ of two prime numbers that satisfy the following conditions!

(1) It holds that $q > p + 1$.

(2) The number $s = p + q$ is also a prime number.

(3) The number $p \cdot q \cdot s$ is divisible by 10.
I. If a pair $(p ; q)$ of prime numbers satisfies the conditions, then:

From (1) and (2), $q$ and $s$ are prime numbers greater than 2, hence odd. However, since $p \cdot q \cdot s$ is even according to (3), $p$ must be even, so $p=2$ (4) must hold. From (3) and (4) it follows:

$q \cdot s$ is divisible by 5. Since $q$ and $s$ are prime numbers, this is only possible if $q=5$ or $s=5$. Due to (2) and (4), this implies $s=7$ or $q=3$.

However, since $q>3$ according to (1) and (4), the only possibility left is $s=7$ and thus $q=5$. Therefore, only the pair $(2 ; 5)$ can satisfy the conditions of the problem.