Prove or disprove $\gcd(q,r) \mid b$ if $a, b, q, r \in \Bbb{Z}^+ \ni a = bq +r$
I'm pretty sure it's true (I can't think of a counter example), but I don't see how to prove it.
Some of my approaches:
$d = \gcd(q,r)$
$d = \gcd(q, a-bq)$
Clearly $d\mid q$ and $d\mid r$ and $d\mid(a-bq)$, but I don't see any way to conclude that $d\mid b$, and approaching it from the other side with $b = \frac{a-r}{q}$ is no help either.