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).


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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.