Prove Proposition (2.19).
Proposition (2.19) is
Let $a = p_1^{e_1} × p_2^{e_2} × ⋯ × p_k^{e_k}$ and $b= p_1^{f_1} × p_2^{f_2} × ⋯ × p_k^{f_k}$ be the decompositions of $a$ and $b$ and $e_j ≥ 0$ and $f_j ≥ 0$. Then the LCM of $a$ and $b$ is given by $[a, b] = p_1^{\max(e_1, f_1)} × p_2^{\max(e_2, f_2)} × ⋯ × p_k^{\max(e_k, f_k)}$.
By definition, the LCM is a multiple of both $a$ and $b$. That is, for some integers $j,k$
$$ [a,b] = ja \quad \land \quad [a,b] = kb $$
That means the primes in the decompositions of $a$ and $b$ must all be in the LCM $[a,b]$.
This means, each prime in $a$ is in $[a,b]$, and each prime in $b$ is in $[a,b]$.
Furthermore, if a prime $p$ occurs $\alpha$ times in $a$, then $p$ occurs in $[a,b]$ at least $\alpha$ times. If it occurred fewer times, then $[a,b]$ would not be a multiple of $a$. Similarly, if a prime $p$ occurs $\beta$ times in $b$, then $p$ occurs at least $\beta$ times in $[a,b]$.
Where a prime $p$ occurs $\alpha \ge 0$ times in $a$ and $\beta \ge 0$ times in $b$, and $\alpha \ne \beta$, then $p$ occurs $\max(\alpha,\beta)$ times in $[a,b]$. If it occurred fewer times in $[a,b]$ then $[a,b]$ would not be a multiple of both $a$ and $b$. If it occurred more frequently in $[a,b]$, then although $[a,b]$ would be a common multiple of $a$ and $b$, it would not be the least common multiple.
This is equivalent to Proposition (2.19).