Assume $a$, $b$, and $c$ are positive integers such that $(a+b+c)$ divides $abc$. Show that $a+b+c$ is composite
I have that so far,
If $a+b+c$ is prime, then letting $a = xd$, $b = yd$, and $c = zd$ we have that $d(x+y+z)$ is prime. This means that $d = 1$ and thus $a,b,c$ are relatively prime. Therefore, $\dfrac{xyz}{x+y+z}$ is supposed to be an integer. What next?