GCD of Integers with Common Divisor

Theorem
Let $a, b \in \Z$ be integers such that not both $a = 0$ and $b = 0$.

Let $k \in \Z_{>0}$ be a strictly positive integer.

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

where $\gcd$ denotes the greatest common divisor.

Proof
Consider the demonstration of the operation of the Euclidean Algorithm applied to $a$ and $b$.

Let each equation be multiplied by $k$.

We have:

This is the operation of the Euclidean Algorithm applied to $k a$ and $k b$.

Hence the greatest common divisor is the last non-zero remainder $r_n k$.

That is:
 * $\gcd \set {k a, k b} = k \gcd \set {a, b}$