# Multiple of Divisor Divides Multiple

## Theorem

Let $a, b, c \in \Z$.

Let:

$a \divides b$

where $\divides$ denotes divisibility.

Then:

$a c \divides b c$

## Proof 1

We have that Integers form Integral Domain.

The result then follows from Multiple of Divisor in Integral Domain Divides Multiple.

$\blacksquare$

## Proof 2

By definition, if $a \divides b$ then $\exists d \in \Z: a d = b$.

Then:

$\paren {a d} c = b c$

that is:

$\paren {a c} d = b c$

which follows because Integer Multiplication is Commutative and Integer Multiplication is Associative.

Hence the result.

$\blacksquare$