Do I understand correctly? How many of the distinct primes $p,q,r$ may divide $p+q+r$? Of course, two of them may, as examples $\{p,q,r\} = \{2,3,5\}$ or $\{p,q,r\} = \{3,5,7\}$ do show. Assume all three; then $pqr \mid p+q+r$, so $pqr \leq p+q+r$, i.e. $\dfrac {1}{pq} + \dfrac {1}{qr} + \dfrac {1}{rp} \geq 1$. But clearly $\min\{pq,qr,rp\} \geq 6$, absurd.