This was a good section, fairly easy to understand. I got a little confused with gcd because we show that the set of all the numbers that are linear combinations of the two numbers in question (i.e. a and b where we're looking for gcd(a,b)) has a least element, even though we're looking for the greatest common denominator. However if we have the greatest common denominator, then the linear combination of the two should be the smallest possible linear combination, or the least element of the set on page 251.
The Euclidean Algorithm is a useful way to find the gcd of really big numbers! The proof was tough to follow, but then the example (example 11.10) laid out actually how to explicitly do it and it made sense. I like the sequence of these sections as well, how we first learned about the division algorithm and then about gcds and now we use all of those in the Euclidean Algorithm.
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