# 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.

### Corollary

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.

## Proof 1

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

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:

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

$\blacksquare$

## Proof 2

 $\ds \exists x, y \in \Z: \,$ $\ds \gcd \set {a k, b k}$ $=$ $\ds \paren {a k} x + \paren {b k} y$ Bézout's Identity: $\gcd \set {a k, b k}$ is the smallest such integer combination $\ds$ $=$ $\ds k \paren {a x + b y}$ $\ds$ $=$ $\ds k \gcd \set {a, b}$ Bézout's Identity

$\blacksquare$

## Examples

### Example: $12$ and $30$

 $\ds \gcd \set {12, 30}$ $=$ $\ds 3 \gcd \set {4, 10}$ $\ds$ $=$ $\ds 3 \times 2 \gcd \set {2, 5}$ $\ds$ $=$ $\ds 6 \times 1$ $\ds$ $=$ $\ds 6$