Solve the equation $p q r = 7(p + q + r)$ in prime numbers.
Since the right-hand side is divisible by 7, one of the numbers is equal to 7. Up to permutation, we can assume that this is $r$. The problem has been reduced to solving the equation $p+q+7=p q$, which can be written as $(p-1)(q-1)=8$.
Since neither of the factors can equal 8 (otherwise the corresponding prime number would be 9), we have $p-1=2, q-1=4$ (or vice versa).
Answer
$\{3,5,7\}$.