Why $15$ divides $z$ (meaning that $z/15$ is an integer): If a prime, in this case $3$, divides a product, in this case $5x$, then the prime must divide one (or both) of the terms. Since $3$ does not divide $5$, we conclude that $3$ must divide $x$.
So $x=3t$ for some $t$, and therefore $x=15t$. We conclude that $15$ divides $z$.
The same kind of reasoning (but easier) shows that $z/5$, $z/3$, and $x/3$ are integers. In fact, we have already shown all these things in the process of showing that $15$ divides $z$. So by a process of elimination, the right answer must be D).
But we can see separately that $z/(xy)$ is not necessarily an integer. For example, let $x=6$, $y=10$, and $z=30$. Then your conditions are met, but $xy$, which is $60$, does not divide $z$.
Note that $z/(xy)$ can be an integer, if $x=3$ and $y=5$. These are in fact the only positive cases when under our conditions, $z/(xy)$ is an integer.