Pairwise Products

Algebra Level 2

$\large xy = 24 \quad xz = 30 \quad yz = 15$

If $x,y,z$ are positive, then $x + y + z$ can be written as $\frac{a\sqrt{b}}{c}$ , where $a,b$ and $c$ are positive integers, with $a$ and $c$ are relatively prime, and $b$ is square-free. Find $a+b+c$ .