Divisibility

is a multiple of , denoted , if there exists a such that . If is a multiple of , we say that is a divisor or factor of , and we say that divides or that is divisible by .

We let .

Let . Then there exists unique such that where .

Proof:
Let . We notice that if we get , so . Thus has a least element with corresponding such that . We have so if then , so . But clearly , so then is not a least element of which is a contradiction. Hence . Since the least element is unique, both and are unique.

For any , if where and , is called the quotient and the remainder.