GCD of Integers with Common Divisor/Proof 1

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.

Let each equation be multiplied by $k$.

We have:

$\ds a k$

$=$

$\ds q_1 \paren {b k} + r_1 k$

where $0 < r_1 k < b_k$

$\ds b k$

$=$

$\ds q_2 \paren {r_1 k} + r_2 k$

where $0 < r_2 k < r_1 k$

$\ds r_1 k$

$=$

$\ds q_3 \paren {r_2 k} + r_3 k$

where $0 < r_3 k < r_2 k$

$\ds \cdots$

$\ds $

$\ds r_{n - 2} k$

$=$

$\ds q_n \paren {r_{n - 1} k} + r_n k$

where $0 < r_n k < r_{n - 1} k$

$\ds r_{n - 1} k$

$=$

$\ds q_{n + 1} \paren {r_n k} + 0$

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:

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

$\blacksquare$

1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.3$ The Euclidean Algorithm: Theorem $2 \text - 7$