Show that if $fd'=f'd $ and the pairs $f, d $ and $f',d' $ are coprime, then $f=f' $ and $d=d' $. Show that if $fd'=f'd $ and the pair $f, d $ are coprime, that is gcd($d,f $)=1, as is the pair $f',d' $, then $f=f' $ and $d=d' $.
This should be simple, but I couldn't verify it.
Thanks in advance!

Hint
Use the Euclid's lemma to show that $f$ divides $f'$ and $f'$ divides $f$ and conclude that $f=f'$ and then $d=d'$.
