1. As I read sections 1.1-1.3, I felt like I understood most of the material. However, for some reason there is one sentence of a theorem that I don't quite understand. I've read it over and over again, but I guess my brain just isn't clicking. It's probably a really stupid question, but what does it mean in Theorem 1.3 when it says, "Furthermore, d is the smallest positive integer that can be written in the form au + bv." I understand that d is the GCD, but I don't get what it means when it says it is the smallest positive integer that can be written in that form. For example, if d is the GCD, then can there be a number greater than d that can be written in the form of that equation? If so, I don't understand that. Also, corresponding with that, since I guess there CAN BE a number other than the GCD that can be written in the form of that equation, then what is it? I bet these are all stupid questions and obvious, but I don't get it right now. Because right after that theorem it gives us a warning that the fact that d= au + bv does NOT imply that d=(a,b) (or in other words is the GCD), then what else can d be? If d is the greatest common divisor between any two numbers, then what number can be greater than d and also satisfy the linear combination equation? Anyway, Sorry if this is confusing, I'm confused. Maybe we can meet and I can better explain what I mean.
2. I found most everything in this chapter to be pretty interesting. I really liked the little short cuts we were taught to make things like finding the GCD a lot easier than finding all the divisors of each number, especially with some of the numbers we were given in the homework including (12378, 3054). That only took like 6 lines to find the GCD. I also found theorem 1.10 interesting. For some reason I had never thought of that before, or for a long while, that ever integer except 0 and +/- 1, is the product of primes. I don't know why I found that so cool but I did. Along with some other vocab words I added to my vocabulary, and a few theorems that I thought were pretty cool, this chapter was very interesting, and I know will help me in my career interests of being a teacher in that I can use these nifty shortcuts to save time and teach others. It was a great chapter to start with! I'm afraid though that there will be a lot more added to the first question as the semester progresses :)
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As an example, look at 6, the GCD of 18 and 30. We can write 6 = 2*30 - 3*18 as an example of d = au + bv. We could also write 12 = 4*18 - 2*30, and get 12 as a combination of 18 and 30. (In fact, we could get any other integer multiple of 6 as a linear combination of 18 and 30, by multiplying both sides of our first equation by an integer.) But 6, the GCD, is the smallest nonzero number we can get in this way; you can never get 3, or 1, or 5.
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