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?

  • 5
    $\begingroup$ Assume without loss of generality that $x \leqslant y \leqslant z$. What is the largest that $x$ can be? For all $x$ not exceeding that bound, what is the largest that $y$ can be? $\endgroup$ – Daniel Fischer Oct 9 '14 at 12:11

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 $$ (x,y,z)=(2,3,6),\ (2,4,4)$$

  • $\begingroup$ what about negative integers? $\endgroup$ – Leox Oct 9 '14 at 12:21
  • $\begingroup$ @Leox "Find all positive integers...". $\endgroup$ – AlexR Oct 9 '14 at 12:21
  • $\begingroup$ Ok, but in the first variant was negative. $\endgroup$ – Leox Oct 9 '14 at 12:25
  • $\begingroup$ In this case x=-y z=1 so there are infinitely many cases $\endgroup$ – HK Lee Oct 9 '14 at 12:39

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