> [!NOTE] Definition (Factor/ Multiple Relation) > For [[Integers|integers]] $a,b\in \mathbb{Z}$, $a|b$ denotes $a$ divides $b$ i.e. $b=ac$ for some $c \in \mathbb{Z}$. We say $a$ is a *factor* or *divisor* of $b$ or $b$ is a *multiple* of $a$. > >For [[Natural Numbers|natural numbers]] $a,b\in\mathbb{N},$ $a$ divides $b$ (or $b$ is a multiple of $a$) iff there exists $c\in \mathbb{N}$ so that $b=ac.$ # Properties The divisibility relation is [[Reflexive Relation|reflexive]], [[Transitive Relation|transitive]] and [[Antisymmetric Relation|antisymmetric]] on $\mathbb{N}$ and $\mathbb{Z}.$ # Applications **Congruence**: Let $a,b, n\in\mathbb{Z}.$ We say that $a$ is [[Congruence Modulo n|congruent]] modulo $n$ to $b$ iff $n|(a-b)$ which is an equivalence relation on $\mathbb{Z}.$ **GCD & LCM**: Given a finite set of positive natural numbers, their [[Greatest Common Divisor (GCD)|GCD]] is defined as the greatest number that divides all of them while their [[Lowest Common Multiple (LCM)|LCM]] is the least number that is a multiple of all of them.