Is it possible for two simplified non integer fractions to have an integer sum and an integer product?
Let’s say the fractions are $\frac{a}{b}$ and $\frac{c}{d}$. For them to multiply to an integer, $d$ must divide $a$ and $b$ must divide $c$. We can express $a$ as $kd$ and $c$ as $mb$ for some integers $m$ and $k$. Their sum is $\dfrac{kd}{b} + \dfrac{mb}{d}$, which simplifies to $\dfrac{kd^2+mb^2}{bd}$. I don’t know how to proceed from here to find if this can ever be an integer. I checked what would happen if $b=d$, but that would mean that the fractions are integers.
