[Math] Math Olympiad Prime Number Question

Question

If $p$, $q$ and $r$ are prime numbers such that their product is $19$
times their sum, find $p^2$ + $q^2$ + $r^2$.

I came across this question in a Math Olympiad Competition and had no idea how to do it. Can anyone help?

Answer 1

One of the primes must be $19$, so WLOG $r=19$. Then $(p-1)(q-1)=20$. There aren't too many ways to factorise $20$...