Honestly there were several difficult portions of the chapter. I didn't understand why the warning for Thm. 1.3 said that d is not necessarily the gcd of a and b since that's exactly what I thought the whole theorem was about. Further, I don't understand how d is the gcd by assumption but it is also the smallest positive integer. Isn't that contradictory? I also had a difficult time understanding the proof that followed. Then I was hung up on the proof of theorem 1.11 where they eliminate a prime on each side until they are left with the sums of the p's equaling 1. I have no idea where the 1 came from. There were a bunch of other places where I only understood most of the proofs so I will probably need to reread everything again.
The most interesting part of the material was the explanation of how Thm 1.1 The Division Algorithm is essentially showing how division is repeated subtraction. We talked about this idea a little in the one math education class we talked about, but it was very interesting to see how this is explained in general terms by proof rather than by examples of oranges or cookies as we did in the other class. I also finally understood prime numbers a little better and Thm 1.10 was really interesting and the proof made a lot of sense to me.
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