# Existence of Greatest Common Divisor/Proof 3

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## 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

From Integers form Integral Domain, we have that $\Z$ is an integral domain.

From Elements of Euclidean Domain have Greatest Common Divisor, $a$ and $b$ have a greatest common divisor $c$.

This proves existence.

From Ring of Integers is Principal Ideal Domain, we have that $\Z$ is a principal ideal domain.

Suppose $c$ and $c'$ are both greatest common divisors of $a$ and $b$.

From Greatest Common Divisors in Principal Ideal Domain are Associates:

- $c \divides c'$

and:

- $c' \divides c$

and the proof is complete.

$\blacksquare$

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $6$: Polynomials and Euclidean Rings: $\S 28$. Highest Common Factor: Example $54$