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

\frac 1a + \frac 1b = \frac 1c

An elegant solution by "completing the rectangle":

&\ \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)

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

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