Euclidean Algorithm/Euclid's Proof/Porism

Porism to Euclidean Algorithm

In the words of Euclid:

From this it is manifest that, if a number measure two numbers, it will also measure their greatest common divisor.

(The Elements: Book $\text{VII}$: Proposition $2$ : Porism)

Proof

An algebraic proof of this is given in Common Divisor Divides GCD.

$\blacksquare$

Historical Note

This proof is Proposition $2$ of Book $\text{VII}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{VII}$. Propositions