The Division Algorithm is a very basic concept, but the proof is crazy. I especially get tripped up when we introduce q and r into the proof. At first I didn't understand the aspect of it being unique. However, I noticed the restriction 0<= r < a, which makes sure that q and r are unique. In the actual proof, why do we consider the set of integers where b-ax>=0? Also why are the integers that satisfy the qualifications for the set postive? It seems that a negative value for x would make sure that b-ax was always greater than zero. I understand that that wouldn't work with our division algorithm, however the inequality and relation for the set don't seem to take this into account.
As we've had types of these problems before, it's neat that we get to learn more about them now. For instance, I thought the application of the division theroem to divisibility of integers (I'm not sure how to say that; where we let a=2,3,4, ... etc and then we know how to write any integer as a product aq+r. For example 2q+1). We learned about this and used it earlier, taking it as true without really learning the proof behind it. Now we know!
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