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 ;-)
Answer to your first question:
This is true by the definition of the rational GCD. Firstly it assures that for your pairs $\left(\dfrac{a}{b}, \dfrac{c}{d}, \dfrac{e}{f} \right)=\left(x, y, z \right)$ we have with $r=\gcd\left(x, y, z \right)$ that $\left(\dfrac{x}{r}, \dfrac{y}{r}, \dfrac{z}{r} \right)$ are all integers and that $r$ is maximal rational number with that property.
Assume $\gcd\left(\dfrac{x}{r}, \dfrac{y}{r}, \dfrac{z}{r} \right)$ is not $1$, then you can multiply $r$ by that number and maintain the property that those numbers are integer. Therefore you get a contradiction to the maximality of $r$.
Answer to your second question:
If the fraction $\dfrac{a}{b}$ is reduced you have that $\gcd(1,\dfrac{a}{b})=\dfrac{1}{b}$ by the above properties. Can you use this to get a formula for $b$?