Farhan's prime problem

All prime numbers are odd, except for $2$ . When two odd numbers are summed, the result is guaranteed to be even. $19$ is odd, and therefore either $5p\space\vdots\space2$ or $3q\space\vdots\space2$ . So either $p$ or $q$ must be $2$ .

If $q$ is $2$ , then: $5p+6=19\implies p=\frac{19-6}{5}=\frac{13}{5} \text{ (F), because }p\text{ must be prime}$ If $p$ is $2$ , then: $10+3q=19\implies q=\frac{19-10}{3}=3$

So the only possible solution is $\boxed{2+3=5}$

The equation 5p+3q=19 implies that one of p,q is even. Since p,q are primes and the only even prime is 2, we have two possibilities. If q=2 then p=13/5 and if p=2 then q=3. So, p+q=5.