It follows from the problem statement that $\frac{ab}{c}$ is a positive integer. Then there should exist positive integers $m, n, x, y$ such that $c = mn$, $a = mx$, $b = ny$. This implies that $d = \frac{ab}{c} = xy$, and so $a+b+c+d = mx+ny+mn+xy = (n+x)(m+y)$, which is, obviously, not prime.