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## Definition: Divisor, Complementary Divisor, Multiple

Divisibility is a relation $R\subseteq \mathbb Z \times \mathbb Z$ denoted by the sign $”\mid”$ and defined as follows:

$$d\mid n:=\Leftrightarrow\exists m\in\mathbb Z\;\; d\cdot m=n\wedge d\neq 0.\label{E18333}\tag{1}$$

In other words, for two integers $n,d\in\mathbb Z$ with $d\neq 0$ $d$ is a divisor of $n$, denoted by $d\mid n$ if and only if there is an $$m\in\mathbb Z$$ with $$dm=n$$. In order to indicate that $$d$$ is a divisor of $$n$$ we write $$d\mid n$$, otherwise we write $$d\not\mid n$$.

There are some related concepts, which shall be introduced here also:

• The integer $$m=\frac nd$$ (if it exists) is unique and called complementary divisor of $$d$$ with respect to $$n$$.
• The number $n$ is called the multiple of $d$ and $m.$1
• A divisor $$d\mid n$$ is called a trivial divisor of $$n$$, if $$d=1$$ or $$d=n$$, otherwise, it is called a non-trivial divisor.
• A proper divisor of $$n$$ is a non-trivial divisor $$d\mid n$$ with $$d\neq n.$$

1 Please note that multiples of $$d=0$$ are undefined. Although $$0\cdot m=0$$ is fulfilled for any $$m$$, we cannot say that $$0\mid 0$$, since $$0$$ is not a divisor of any number (because by definition $\ref{E18333},$ it cannot be for $d=0$). We want to have a definition of multiples which is complementary to the definition of divisors.