Sum of Unit Fractions

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

#NumberTheory

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

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:
$(-1,-1),(1,1),(-2,-2),(2,2)$ .

Jesse Nieminen - 4 years, 10 months ago

Ayanlaja Adebola - 4 years, 10 months ago

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

Rex Holmes - 4 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$