GCD with Remainder

Theorem
Let $a, b \in \Z$.

Let $q, r \in \Z$ such that $a = q b + r$.

Then $\gcd \left\{{a, b}\right\} = \gcd \left\{{b, r}\right\}$

where $\gcd \left\{{a, b}\right\}$ is the greatest common divisor of $a$ and $b$.

Proof
The argument works the other way about:

Thus $\gcd \left\{{a, b}\right\} = \gcd \left\{{b, r}\right\}$.