Show $\gcd (a,b)=\gcd (b,r)$ if $a = bq + r$

Question

Let $a, b$ be two integers with $b \neq 0$, and $q, r$ non-negative integers such that $a = bq + r$. How can we show that $\gcd (a,b)=\gcd (b,r)$?

Answer 1

If $x$ divides $a$ and $b$, then $x$ divides $a-bq=r$. If $y$ divides $b$ and $r$, then $y$ divides $bq+r=a$.

Answer 2

Hint $\ $ If $\ d\mid b\ $ then $\ d\mid \color{#0a0}{bq+r}\iff d\mid \color{#c00}r.\ $ Thus $\ \color{#0a0}{bq+r},b\ $ and $\ \color{#c00}r,b\ $ have the same set $S$ of common divisors $\,d,\,$ so they have the same greatest common divisor $\,(= \max S).$