2.3, due January 14

1. Difficult: There was a part of the proof of corollary 2.9 that I didn't understand. When it's proving existence it says that a(w-ub)=aw-aub=b-b=0 but this seems a little presumptuous. Because it stats thta x-ub, so this really says that a(w-x)=aw-ax and we already supposed that w=x, so why isn't it that it's a(0)=b? I'm so confused.
2. Reflective: I thought it was very interesting that for non-prime equivalence classes, if [a][b]=0, then it doesn't mean that [a]=0 or [b]=0 BUT for a prime equivalence class if [a][b]=0 then [a] or [b] DOES =0. It took me a minute to wrap my head around that, but it makes perfect sense. Because the only time you have [a][b]=0 is when [a] and [b] are the two parts of the factorization of n (in mod n). But since primes have no factorization, the only way to get 0 is to multiply by zero! It makes perfect sense but I had never thought of it before.