Let $p$ and $q$ be distinct primes. Show that $[p, q] = p × q$.
The prime decompositions of $p$ and $q$ are as follows.
$p = p^1 \times q^0$
$q = p^0 \times q^1$
The LCM, using Proposition (2.19), is
$$\begin{align} [p,q] &= p^{\max(1,0)} \times q^{\max(0,1)} \\ \\ &= p^1 \times q^1 \\ \\ &= p \times q \end{align}$$
An alternative approach is to use Proposition (2.20)
Let $a$ and $b$ be relatively prime then $[a, b]$ = $a × b$.
Since $p$ and $q$ are distinct primes, they are coprime, and so the result follows immediately.