OK, I've just realised (2) has nothing to do with congruences and just comes from manipulation of powers. I still don't get (1).
I am having some trouble learning how to manipulate congruences, and have two questions on things I don't understand:
1. I understand that the following:
comes directly from the definition of the rule for
But why is the following also true?:
2. Why is the following true?
Remember this is just addition so you don't need those brackets :
By simplifying :
Which must be true.
Basically, in a congruence you can do about everything except using logarithms, you need care when working with square roots, and to divide you must multiply by the inverse modulo n. Here are the formulae :
if and only if .
This is about everything to begin. You'll see that we can also simplify powers using various powerful tools.