Solve $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$
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Lecture Notes

This is the first pattern which can be solved using the factorization method: $$\frac{1}{x} + \frac{1}{y} = \frac{1}{n}\Leftrightarrow (x-n)(y-n)=n^2$$

Alternatively, such an equation can also be solved by noting $$\frac{1}{x} + \frac{1}{y} \Leftrightarrow x = n +\frac{n^2}{y-n}$$

Then, because $x$ is an integer, $\frac{n^2}{y-n}$ must be an integer too. It follows that $(y-n)$ must be a divisor of $n^2$.

We may note that in this case, the two methods described above are essentially the same.