Divide

To divide is to perform the operation of Division, i.e., to see how many times a Divisor $d$ goes into another number $n$. $n$ divided by $d$ is written $n/d$ or $n\div d$. The result need not be an Integer, but if it is, some additional terminology is used. $d\vert n$ is read ``$d$ divides $n$'' and means that $d$ is a Proper Divisor of $n$. In this case, $n$ is said to be Divisible by $d$. Clearly, $1\vert n$ and $n\vert n$. By convention, $n\vert$ for every $n$ except 0 (Hardy and Wright 1979). The ``divided'' operation satisfies

$$b\vert a{\rm\ and\ }c\vert b \Rightarrow c\vert a$$

$$b\vert a \Rightarrow bc\vert ac$$

$$c\vert a{\rm\ and\ }c\vert b \Rightarrow c\vert(ma+nb).$$

$d'\notdiv n$ is read ``$d'$ does not divide $n$'' and means that $d'$ is not a Proper Divisor of $n$. $a^k \vert\vert b$ means $a^k$ divides $b$ exactly.

See also Congruence, Divisible, Division, Divisor

References

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 1, 1979.