In favour of the Euclidean Algorithm

Mathematics teachers often talk as though algorithms are boring – we prefer to call our methods ‘procedures’ or ‘processes’, because ‘algorithms’ sound like rote learning, and something we should shun. But algorithms are often beautiful (see Fry, 2018), and the Euclidean Algorithm is a good example of this. I am not sure why it is not taught more frequently in schools, and why, when it is, it tends (at least in my experience) to be introduced as an ‘extension’ for higher attainers. It is often a very efficient method. [...]