I tried to prove this theorem, but I'm really confused about its correctness.
Let's say
$\gcd( a, b ) = 1 \Rightarrow am + bn = 1 \text{ where } m, n\in\mathbb Z$
Then by multiplying both side by $c$, we have: $amc + bnc = c$. From here, is it sufficient to say $c = \gcd( ac, bc )$?
I feel like this proof needs some more explanation, but I can't think of one. Any idea?
Thanks,