<s>[INST] Find all integers $x$, $y$, and $z$ such that $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$ Characterize all positive integers $x$, $y$, and $z$ such that:
$$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{z}$$
For example, $\dfrac{1}{x+1} + \dfrac{1}{x(x+1)} =  \dfrac{1}{x}$.
 [/INST] ${1\over x}+{1\over y}={1\over z}$ if and only if it has the form
$${1\over a(a+b)d}+{1\over b(a+b)d}={1\over abd}\quad\text{with}\quad \gcd(a,b)=1$$
(all variables being understood as positive integers).
The "if" part is trivial to verify.  The "only if" part comes by setting $x=ga$ and $y=gb$ with $g=\gcd(x,y)$, which gives
$${1\over x}+{1\over y}={a+b\over gab}$$
and noting that $a+b$ is relatively prime to both $a$ and $b$.
</s>