2.4.2, due on September 9

I feel like there was a simpler way of finding multiplicative inverses in Z modulo n, but I can't remember what it was. The euclidean algorithm and writing 1 as a linear combination of a and n makes sense, I just thought I remembered a shorter way of finding these inverses. I do feel like after reading the proof again, I better understand the process of finding multiplicative inverses and why it works. With Wilson's Theorem, I understood about the first half and then it seemed like they dropped one proof and started another so I got confused. I also didn't understand why Wilson's theorem is useful.