# discrete mathematics – Struggling to understand basics of complete residue system

I’m really struggling to understand the literal arithmetic being applied to find a complete residue system of modulo $$n$$. Below is the definition my textbook provides along with an example.

Let $$k$$ and $$n$$ be natural numbers. A set $${a_1,a_2,…,a_k}$$ is called a canonical complete residue system modulo $$n$$ if every integer is congruent modulo $$n$$ to exactly one element of the set

I’m struggling to understand how to interpret this definition. Two integers, $$a$$ and $$b$$, are “congruent modulo $$n$$” if they have the same remainder when divided by $$n$$. So the set $${a_1,a_2,…,a_k}$$ would be all integers that share a quotient with $$b$$ divided by $$n$$?

After I understand the definition, this is a simple example provided by my textbook

Find three residue systems modulo $$4$$: the canonical complete residue system, one containing negative numbers, and one containing no two consecutive numbers

My first point of confusion is “modulo $$4$$“. $$a{space}mod{space}n$$ is the remainder of Euclidean division of $$a$$ by $$n$$. So what is meant by simply “modulo $$4$$“? What literal arithmetic do I perform to find a complete residue system using “modulo $$4$$“?