Division is defined between natural numbers. However there are many cases where this division does not map to a natural number. For example:
\(\dfrac{7}{3}\)
We can divide \(6\) of the \(7\) by \(3\), giving \(2\) with \(1\) remaining.
Alternatively we can divide \(3\) of the \(7\) by \(3\), giving \(1\) with \(4\) remaining
Or we could divide \(0\) of the \(7\) by \(3\) giving \(0\) with \(7\) remaining.
The remainder refers to the lowest possible number - in this case \(1\).
This is the set of numbers from \(0\) to \(n-1\).
This a set of numbers none of which are congruent \(\mod n\). That is, for no pair \(\{a,b\}\) does \(a \mod(n)=b mod(n)\)
This is a complete residue system where all numbers are relatively prime to \(n\).
\(5\) and \(11\) are congrument \(\mod 3\)
If \(a \mod(n)=b mod(n)\) then \(a\) and \(b\) are congruent mod \(n\).