find the number of ordered triples $(x,y,z)$ satisfying $$x\mid yz-1$$ $$y\mid xz-1$$ $$z\mid xy-1$$ and $0<x,y,z<2014$
note: $x,y,z$ are integers
I found $(2,3,5)$ as one set of solution (actually $6$), and I tried change $$x\mid yz-1\Longrightarrow xk_1=yz-1$$but it didn't work.
Any idea/hint of how can I make progress?
Thank you in advance.
Note:I DO NOT want a computer testing all combinations.