How find this $xyz|(xy-1)(xz-1)(yz-1)$
Your calculation is a useful beginning. You have concluded that $xyz$ divides $xy+yz+zx-1$. Let us suppose that $x\gt y\gt z$. Note that $xy+yz+zx-1\lt 3xy$. So $z=1$ or $z=2$. But it cannot be $1$.
We have reached the equation $xy+2(x+y)-1=2xy$. The rest is left to you.