**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.

| | | | | created: 2014-06-21 15:13:20 | modified: 2019-04-17 07:19:55 | by: *bookofproofs* | references: [696]

## 1.**Example**: Divisibility Examples

[696] **Kramer Jürg, von Pippich, Anna-Maria**: “Von den natürlichen Zahlen zu den Quaternionen”, Springer-Spektrum, 2013

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