Integers

Set of all whole numbers (incl. 0, negative): $z=\{0,\pm 1,\pm 2,\pm 3,...\}$

Divisibility on $z$ §

$a,bϵz$ : $b$ divides $a$ if there is $cϵz$ such that $a=b\times c$. This is written as $b|a$, e.g. $3|6\because 3\times 2=6$ . In the same way, we can say something does not divide by crossing out the $|$.

$b|a⟺\frac{a}{b}ϵz$ : b divides a if and only if a divided by b exists in the set of integers (is an integer). For example, if $0|a:a\ne 0$, there are no solutions. But, if $a|0$ then there exists a solution, $c=0\because 0=a\times c⟹0=a\times 0$.

For $aϵz$ we define $D(a)=\{xϵz:x|a\}$, the set of divisors of $a$. For example, $D(6)=\{\pm 1,\pm 2,\pm 3,\pm 6\}$. This is the same set as $-a$, e.g. $D(6)=\{\pm 1,\pm 2,\pm 3,\pm 6\}=D(-6)$. So we may propose a statement that $D(a)=D(-a)$ for any $aϵz$.

Another set may be $D(0):D(0)=z$, a rather unique set. This is because $0|0$ is defined as true since there always exists a solution for $0=0\times c$.

Example $D(24)=\{\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 8,\pm 12,\pm 24\}\because 24=2^3\times 3$; finding a Divisibility set using Prime factorisation.