<s>[INST] Find all positive integers $(x,y,z)$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ is integer As stated in title, I would like to find solution to this problem:

Find all positive integers $(x,y,z)$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ is also integer.

I need idea how to solve this?
 [/INST] Let $1\leq x\leq y\leq z$ and $N= \frac{1}{x} + \frac{1}{y} +
\frac{1}{z}$
If $N=3$ then $$x=y=z=1$$
If $N=2$ then $$ x=1,\ y=z=2$$
If $N=1$ then $$ 2\leq x\leq 3 $$
If $x=3$ then $$ (x,y,z)=(3,3,3) $$
 If $x=2$ then $$ 3\leq y \leq 4$$
So $$ (x,y,z)=(2,3,6),\ (2,4,4)$$
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