Prove or disprove: (i) gcd(a,q) = gcd(q,r) (ii) gcd(q,r)|b (iii) gcd(b,r) = gcd(a,q) (iv) gcd(a,r)|q

Question

Given $a,b,q,r \in ℤ \ni a = bq + r$. Prove or disprove the following:

(i) $\gcd(a,q) = \gcd(q,r)$

(ii) $\gcd(q,r)|b$

(iii) $\gcd(b,r) = \gcd(a,q)$

(iv) $\gcd(a,r)|q$

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).

Answer 1

Let $g = \text{gcd}(q,r)$. We know that $g= \text{gcd}(a,q)$, so $g$ divides $a$ and $q$ and $r$. But there is no way to prove that $g$ divides $b$. This is indeed not always the case. But if you have a counterexample, then it's okay, right?