Solve $\frac{1}{x}+\frac{1}{y}=\frac{m}{n}$ Intermediate

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Lecture Notes

This pattern, though similar to Solve $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$, usually cannot be solved using the factorization method. A typical solving technique is the squeeze method by assuming $x \le y$ (the symmetry property): $$x \le y \implies \frac{1}{x} \ge \frac{1}{y} \implies \frac{1}{x} \ge \frac{1}{2}\cdot\frac{m}{n} \implies x \le \frac{2n}{m} $$

Then, we can check every positive integer satisfying this condition to see whether the correspoding $y$ is an integer. If it is, then such a pair is a solution.