Definition 1: (divides, divisor, multiple)
Let
, with
. We say that "
divides
" or that "
is a multiple of
", if there exists some
such that
.
We write this as
.
Proposition 1: (some elementary properties of division)
Let
be integers. Then
- If
and
, then
. ▶
□
- If
and
, then
.
- If
and
, then
.
- If
and
, then
. ▶
□
Examples:
because
. However
: if it did, would also divide
(by Proposition 1, point 3), which is impossible (Proposition 1, point 1). Similarly,
.
Proposition 2: (division algorithm)
Let
, with
. Then
, for some
, with
.
▶