The Set of Integers Modulo n

Abstract Algebra

Congruent Modulo n

Let n be greater than or equal to 2. Then a and b are congruent modulo n if n|a-b. If so, we write a (triple equals) b (mod n).

Divisible Theorem

n in Z is divisible by 9 if and only if the sum of its digits is divisible by 9.

Equivalence Relation

A relation R on a set S is an equivalence relation if the following three properties hold:

(1) Reflexive: for all a in S, a~a

(2) Symmetric: if a~b, the b~a

(3) Transitive: if a~b and b~c, then a~

Congruence Theorem

Congruence modulo n is an equivalence relation on Z.

Equivalence Class

An equivalence relation R on a set S partitions S into disjoint pieces S_i such that S=S₁US₂U… Each S_i is called an equivalence class.

Residue Class Modulo n

If a in Z, then its equivalence class [a] with respect to congruence modulo n is called its residue class modulo n & we write a for convenience. a={x in X|x (triple equals) a (mod n)}.

Theorem 3

Given n is greater than or equal to 2, a=b ←→a (triple equals) b (mod n).

Set of integers modulo n

The set of integers modulo n is denoted Z_n and is given by Z_n={0,1, 2,…n-1}.

Group of Units

The group of units in Z_n. Let Z_n^x be the set of elements in Z_n which it has a multiplicative inverse.