**Hint** $ $ note if $\rm\, d\mid \color{#c00}b\ $ then $\rm\ d\,\mid\, q \color{#c00}b + r \!\iff d\ |\ r, \ $ i.e. *arithmetically* in congruence language:

$\!\rm\bmod d^{\phantom{|^|}}\!\!\!\!:\ $ if $\rm\ \color{#c00}{b\equiv 0}\ $ then $\rm\ q\color{#c00}b+r\equiv 0 \!\iff\! r\equiv 0\ $ by congruence sum & product rules.

By above $\rm\, \{qb+r,b\}\, $ and $\rm\, \{r,\, b\}\, $ have the same set $\,\rm S\,$ of common divisors $\rm\,d,\,$ which implies that they also have the same *greatest* common divisor $\rm(= \max S)$.

Thus $\rm \,\bbox[5px,border:1px solid #c00]{\gcd(a,b) = \gcd(r,b)\,\ {\rm if}\,\ a\equiv r\pmod{\! b}} \,$ since then $\rm\, a = qb+r\,$ for $\,r\in\Bbb Z$

e.g. $\ \ \rm \bbox[5px,border:1px solid #c00]{\gcd(a,b) = \gcd(a\bmod b,b)}\,\ $ by choosing $\rm \, r = a\bmod b,\, $ using the *division algorithm*. $ $ This **gcd modular reduction** is the descent (induction) step in the well-known classical recursive Euclidean algorithm for the gcd.

Also $\rm\, d\mid a\iff d\mid r\ $ **if** $\rm \ a\equiv r\pmod{\! d},\,$ since then $\rm\,a = qd+r\,$ so the above applies.

So $\ \ \,\rm \bbox[5px,border:1px solid #c00]{\!d\mid a\iff d\mid (a\bmod d)}.\,$ This **divisibility mod reduction** often simplifies matters.

Generally the set of multiples of $\rm\,d\,$ are closed under integral linear combinations, and ditto for *common* multiples of any set of integers, which leads to the universal property of the lcm, using descent via the Euclidean division algorithm.

**Remark** $ $ The result holds true because $\rm\,\mathbb Z\,$ forms a *subring* of its fraction field $\rm\,\mathbb Q.\,$ More generally, given any subring $\rm\,Z\,$ of a field $\rm\,F\,$ we define divisibility relative to $\rm\ Z\ $ by $\rm\ x\mid y \iff y/x\in Z.\,$ The above proof still works, since if $\rm\ q,\ b/d\ \in Z\, $ then $\rm\, q\,(b/d) + r/d\in Z\iff r/d\in Z.\,$ Thus the common divisibility laws follow from the fact that (sub)rings are closed under subtraction and multiplication (subring test). Being so closed, $\rm\,Z\,$ serves as a ring of "integers" for divisibility tests.

For example, to focus on the prime $2$ we can ignore all odd primes and define a divisibility relation so that $\rm\ m\ |\ n\ $ if the power of $2$ in $\rm\,m\,$ is $\le$ that in $\rm\,n\,$ or, equivalently if $\rm\ n/m\ $ has odd denominator in lowest terms. The set of all such fractions forms a ring $\rm\,Z\,$ of $2$-integral fractions. Moreover, this ring enjoys parity, so arguments based upon even/odd arithmetic go through. Similar ideas lead to powerful *local-global* techniques of reducing divisibility problems from complicated "global" rings to simpler "local" rings, where divisibility is decided by simply comparing powers of a prime.