Let $a$, $b$ and $a$, $c$ be relatively prime positive integers. Prove that $[a, bc] = a × b × c$.
Hint: You may find the result of Exercises 1.3, question 15 (i) useful:
$$\gcd (a, b) = \gcd (a, c) = 1 ⇔ \gcd (a, bc) = 1$$
We'll use Proposition (2.20)
Let $a$ and $b$ be relatively prime then $[a, b] = a × b$.
We are given $\gcd(a,b)=1$ and $\gcd(a,c)=1$. By Exercise (1.3).15(i), this means $\gcd(a,bc)=1$.
Since $a$ and $bc$ are coprime, Proposition (2.20) immediately gives us
$$ [a,bc] = a \times bc = a \times b \times c $$