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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 ;-)

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