Find all prime numbers $p$ and $q$, such that $7p+q$ and $pq+11$ are
also prime numbers.
Based on the fact that all primes, besides 2, are odd, I found that either $p$ or $q$ must be $2$ in order for $pq+11$ to be a prime number. From here, I found several pairs of $p$ and $q$ that work, but I don't know how to find all $p$ and $q$. I tried letting
$7p+q=r$
$pq+11=s$
and then adding the equations and using SFFT to get:
$(p+1)(q+7)=r+s-4$
but it doesn't really help.
Best Answer
So, say $p=2$. Then your expressions are $14+q$ and $2q+11$. Working $\pmod 3$ we see that these are $q-1$ and $2-q$ Easy to see that one of these is divisible by $3$ unless $q=3$ which is a valid example.
Now say $q=2$, Then your expressions are $7p+2$ and $2p+11$. Working $\pmod 3$ we see that these are $p+2$ and $2(p+1)$ and again one of these terms must be divisible by $3$ unless $p=3$, which is again a valid example.