Product of Divisors is Divisor of Product

Theorem
Let $a, b, c, d \in \Z$ be integers such that $a, c \ne 0$.

Let $a \divides b$ and $c \divides d$, where $\divides$ denotes divisibility.

Then:
 * $a c \divides b d$

Proof
By definition of divisibility:

Then:

and the result follows by definition of divisibility.