Find all positive integer pairs \( a,b \) such that for given positive integer \( c \)

\begin{equation}
\frac 1a + \frac 1b = \frac 1c
\end{equation}

An elegant solution by "completing the rectangle":

\begin{alignat}{3}
&\ \frac 1a + \frac 1b &=&\ \frac 1c
\\\iff&\ c(a+b) &=&\ ab
\\\iff&\ c^2 &=&\ ab - c(a+b) + c^2
\\\iff&\ c^2 &=&\ (a-c)(b-c)
\end{alignat}

Now just list inspect the divisors of \(c^2\).