# Existence of Greatest Common Divisor/Proof 2

## Theorem

Let $a, b \in \Z$ be integers such that $a \ne 0$ or $b \ne 0$.

Then the greatest common divisor of $a$ and $b$ exists.

## Proof

By definition of greatest common divisor, we aim to show that there exists $c \in \Z_{>0}$ such that:

\(\ds c\) | \(\divides\) | \(\ds a\) | ||||||||||||

\(\ds c\) | \(\divides\) | \(\ds b\) |

and:

- $d \divides a, d \divides b \implies d \divides c$

Consider the set $S$:

- $S = \set {s \in \Z_0: \exists x, y \in \Z: s = a x + b y}$

$S$ is not empty, because by setting $x = 1$ and $y = 0$ we have at least that $a \in S$.

From the Well-Ordering Principle, there exists a smallest $c \in S$.

So, by definition, we have $c > 0$ is the smallest such that $c = a x + b y$ for some $x, y \in \Z$.

Let $d$ be such that $d \divides a$ and $d \divides b$.

Then from Common Divisor Divides Integer Combination:

- $d \divides a x + b y$

That is:

- $d \divides c$

We have that:

\(\, \ds \exists t, u \in \Z: \, \) | \(\ds a\) | \(=\) | \(\ds c t + u:\) | \(\ds 0 \le u < c\) | Division Theorem | |||||||||

\(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds a x t + b y t + u\) | Definition of $c$ | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds u\) | \(=\) | \(\ds r \paren {1 - x t} + b \paren {-y t}\) | rearranging | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds u\) | \(=\) | \(\ds 0\) | as $u < c$ and the definition of $c$ | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds c\) | \(\divides\) | \(\ds a\) | Definition of Divisor of Integer |

- $c \divides b$

Now suppose $c'$ is such that:

\(\ds c'\) | \(\divides\) | \(\ds a\) | ||||||||||||

\(\ds c'\) | \(\divides\) | \(\ds b\) |

and:

- $d \divides a, d \divides b \implies d \divides c'$

Then we have immediately that:

- $c' \divides c$

and by the same coin: $c \divides c'$

and so:

- $c = c'$

demonstrating that $c$ is unique.

$\blacksquare$

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*... (previous) ... (next): Introduction $\S 6$: The division process in $I$ - 1994: H.E. Rose:
*A Course in Number Theory*(2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization: Theorem $1.2$ - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $3$: The Integers: $\S 11$. Highest Common Factor: Theorem $19$