Disclaimer: I'm an engineer, not a mathematician
I have a set of three fractions (a/b, c/d, e/f). I can multiply them all by another fraction, so that their mutual ratios remain the same. I want to end with natural numbers (i, j, k) where
$\gcd(i, j, k) = 1$
I tried the following:
$ \dfrac{g}{h} = \gcd\left(\dfrac{a}{b}, \dfrac{c}{d}, \dfrac{e}{f} \right) $
Then
$ \begin{cases} i = \dfrac{a \cdot h}{b \cdot g} \\ \\ j = \dfrac{c \cdot h}{d \cdot g} \\ \\ k = \dfrac{e \cdot h}{f \cdot g} \end{cases} $
seems to work, but I can't prove it's always true. Is this a valid conjecture?
Another problem I ran into: I needed the denominator of a reduced fraction, and I couldn't find it! There sure must be a function $f$ where
$f\left(\dfrac{a}{b}\right) = b$
for the reduced fraction $\dfrac{a}{b}$?
I'm not a mathematician, so please type slowly ;-)