Regarding the value of $X+Y+Z$ if $\frac1X + \frac1Y + \frac1Z = 1$

Question

Given $X, Y$ and $Z$ are three positive and non-equal natural numbers. If $\frac1X + \frac1Y + \frac1Z = 1$, then what's the value of $X + Y + Z$?

This is a question from my nephew, I cannot think of any such triples. Any idea guys?

Answer 1

There are a lot of possible solutions. A simple calculation shows that $x,y,z$ need to satisfy $$xy\neq 0,$$ $$xy-x-y\neq 0,$$ $$z=\frac{xy}{xy-x-y}.$$ It is now simple to produce solutions. For example set $x=2$. Then you obtain $y\neq 0,2$ and $z=\frac{2y}{y-2}$ which yields for example the solution $(x,y,z)=(2,3,6)$ and its permutations. Here the sum of the numbers is eleven. I was letting a computer do the calculations up a certain bound and it seems to me that this is the only solutions with $x,y,z$ mutually distinct.

EDIT: According to the post linked in the comments, these are indeed the only solutions with mutually distinct $x,y,z$. So I guess the answer your nephew was looking for is that $x+y+z=11$.