GCD of Integers with Common Divisor/Corollary

Theorem

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

Let $k \in \Z_{\ne 0}$ be a non-zero integer.

Then:

$\gcd \set {k a, k b} = \size k \gcd \set {a, b}$

where $\gcd$ denotes the greatest common divisor.

From GCD of Integers with Common Divisor the case where $k > 0$ has been demonstrated.

It remains to demonstrate the case where $k < 0$.

Indeed:

$-k = \size k > 0$

and so:

$\ds \gcd \set {a k, b k}$

$=$

$\ds \gcd \set {-a k, -b k}$

$\ds $

$=$

$\ds \gcd \set {a \size k, b \size k}$

$\ds $

$=$

$\ds \size k \gcd \set {a, b}$

$\blacksquare$

1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.3$ The Euclidean Algorithm