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Question:

let $x,y,z$ be postive integers and $x\neq y\neq z$ such $$xyz|(xy-1)(xz-1)(yz-1)$$ Find all $x,y,z$,

my try: $$(xy-1)(yz-1)(xz-1)=x^2y^2z^2-xyz(x+y+z)+(xy+yz+xz)-1$$ then $$\dfrac{(xy-1)(yz-1)(xz-1)}{xyz}=xyz-(x+y+z)+\dfrac{xy+zy+xz-1}{xyz}$$ so $$\dfrac{xy+yz+xz-1}{xyz}\in N$$ Then I can't get any further

I found $(x,y,z)=(2,3,5),(2,5,3),(3,2,5),(3,5,2),(5,2,3),(5,3,2)$such it, I don't know/have any other,

also How to prove it.

Thank you very much

Alfons Marklén
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1 Answers1

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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.

André Nicolas
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