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:
Examples: 1
Corollaries: 1 2
Definitions: 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Examples: 18 19 20 21
Explanations: 22
Lemmas: 23 24 25 26
Motivations: 27
Parts: 28 29
Proofs: 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74
Propositions: 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98
Theorems: 99 100 101 102
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. ↩