Recitation of Apr. 11, 2013

We discussed existence of multiplicative inverses in modular arithmetic, and solved problem 7.1 . A few examples of equations of the form ax ≡ b mod n were considered, and we finally talked about the general case. We quickly looked at Problem 7.20 which gives a neat proof that k - 1 divides kn - 1.

In my opinion, the most important facts about modular arithmetic we have seen so far are:

  1. For all relatively prime integers a and b, there are integers s and t such that sa + tb = 1.
  2. Multiplicative inverses do not always exist.

The first fact is the one from which pretty much everything else you have seen can be derived, and the second is an important warning.