GCD with Remainder

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

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

Then:
 * $\gcd \set {a, b} = \gcd \set {b, r}$

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

Proof
The argument works the other way about:

Thus:
 * $\gcd \set {a, b} = \gcd \set {b, r}$