The integers modulo n
Formal definition of integers mod n
Definition: Let \(n\) be a natural number greater than 1. The set of integers modulo \(n\), written \(\mathbb{Z}_{n}\), is the set of equivalence classes \([a]\) for the equivalence relation defined by congruence modulo \(n\).
Remark: The book gives a careful walkthrough of an example in the case where \(n=5\).
Properties of \(\mathbb{Z}_{n}\)
Proposition: \(\mathbb{Z}_{n}\) has \(n\) elements \(\{[0],[1],\ldots, [n-1]\}\).
Arithmetic in \(\mathbb{Z}_{n}\)
Proposition: Define \([a]+[b]=[a+b]\) and \([a][b]=[ab]\). Then these are well-defined operations, meaning that if \([a]=[a']\) and \([b]=[b']\) then \([a]+[b]=[a']+[b']\), and similarly for multiplication.