Multiple of Divisor Divides Multiple/Proof 2

Theorem

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

Let:

$a \divides b$

where $\divides$ denotes divisibility.

Then:

$a c \divides b c$

Proof

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.

$\blacksquare$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $3$: The Integers: $\S 10$. Divisibility: Theorem $16 \ \text{(ii)}$