Sum of Unit Fractions

For which values of $$A$$ and $$B$$, the number $$\dfrac1A + \dfrac1B$$ is an integer?

Note by Lucas Nascimento
2 years ago

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Let's find all possible solutions for $$A\ \text{and}\ B$$.

$$\dfrac1A+\dfrac1B = \dfrac{A+B}{AB}$$ which implies that the number is an integer iff $$AB\ |\ A+B$$.

Clearly all pairs of solutions where $$A = -B \neq 0$$ are valid.

Now if $$A+B \neq 0$$, $$AB\ |\ A+B$$ implies that $$A \ |\ A+B\implies A \ |\ B\ \text{and}\ B \ |\ A+B\implies B \ |\ A$$ which implies that $$A=B$$.

Now $$A^2 \ |\ 2A\implies A \ |\ 2$$.

Using the facts above, we know that the only possible pairs of solutions $$(A,B)$$ (when $$A+B \neq 0$$ ) are:
$$(-1,-1),(1,1),(-2,-2),(2,2)$$.

- 2 years ago

I know possible positive values a (1, 1) and (2,2)

- 2 years ago

1

- 2 years ago