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 k^{n} - 1.

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

- For all relatively prime integers a and b, there are integers s and t such that sa + tb = 1.
- 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.