Hi Howard ,
Have you seen this article written in Khan Academy about Euclid's Algorithm?
https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic/a/the-euclidean-algorithm

It may explain some of your questions in the topic your looking .Also should you have any further questions I will be happy to answer them for you.
Proud to be a Khan Academer
Katrina

I found an article about Euclid's Algorithm, but I think it would be better if it was either condensed, done into a video, or inserted into the Pre-algebra section for Factors and Multiples. I believe it would be easier if all three were done or the article was made into a video and put into the math section. 

Apple Wu

Redapple, is the article you are thinking about this one? If so it's not really integrated with the math section, it's part of the Computer algorithm theme. The Computer algorithm theme is meant for older students than the https://www.khanacademy.org/math/pre-algebra/pre-algebra-factors-multiples/pre-algebra-greatest-common-divisor/v/greatest-common-divisor video.

However, for students who are interested in alternative methods, the discussion under the video mentions both Euclid's Algorithm and the fraction method mentioned by Byron Lawson could be of interest.