Number theory is concerned with the properties of the integers. One of the most important is divisibility.
Let and be integers with We say that divides if there is an integer such that This is denoted by Another way to express this is that is a multiple of
(does not divide).
The following properties of divisibility are useful.
Let represent integers.
For every and Also, for every
If and then
If and then for all integers and
Proof. Since we may take in the definition to obtain Since we take to prove Since