Given $a,b,q,r \in ℤ \ni a = bq + r$. Prove or disprove the following:
(i) $\gcd(a,q) = \gcd(q,r)$
(iii) $\gcd(b,r) = \gcd(a,q)$
Part (i) is no problem. I'm getting hung up on part (ii). After doing some examples I can see that gcd(q,r) does not always divide b. How do I approach disproving the statement? I feel like I'm missing something simple here. The rest should follow if I can manage part (ii).