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# number theory

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Given that x, 2/x, y, 2/y, z, and 2/z are all integers, how many distinct values of x + y + z are possible?

Apr 5, 2021

Given that x,y, and z are all integers, we have $\frac{2}{x}=1$ or $2$. So we have $x=1$ or $x=2$. Symmetrically, the same is true for $y$ and $z$. So we have $1+1+1$, $2+1+1$, $2+2+1$, and $2+2+2$ as our only distinct options. Counting, we see there are $\boxed{4}$ of them.