Show that $[ab, ac] = a × [b, c]$ where $a$, $b$, and $c$ are positive integers.
Hint: You may find the following Proposition (1.11) helpful: $\gcd (ab, ac) = a × \gcd (b, c)$ provided $a > 0$.
We'll use Proposition (2.22).
Let $a$ and $b$ be positive integers then $\gcd (a, b) × [a, b] = a × b$.
In our scenario, the proposition tells us
$$ ab \times ac = \gcd(ab, ac) \times [ab, ac] $$
That is
$$[ab, ac] = \frac{ab \times ac}{\gcd(ab, ac)}$$
Using Proposition (1.11)
$$\begin{align} [ab, ac] &= \frac{ab \times ac}{a \times \gcd(b, c)} \\ \\ &= a \times \frac{b \times c}{ \gcd(b, c)} \\ \\ & = a \times [b,c] \end{align}$$