Why does prime factorization for LCM work?

A method to finding the LCM of two numbers is to use prime factoring. I know how to do this, but logically why does this work in finding the lcm

By FTA (existence and uniqueness of prime factorizations), divisibility reduces to divisibility in each prime component, i.e. $\ p^{\large b} q^{\large b'}\!\cdots\mid p^{\large a} q^{\large a'}\!\cdots\!\iff p^{\large b}\mid p^{\large a}\ $ & $\,\ q^{\large b'}\!\mid q^{\large a'}\ \ldots\ $

So $\,\ B,C\mid A\iff p^{\large b},p^{\large c}\mid p^{\large a}\ $ & $\,\ q^{\large b'},q^{\large c'}\mid q^{\large a'}\ \ldots\ $ But we have

$$\qquad\qquad\ \, p^{\large b},p^{\large c}\mid p^{\large a}\! \iff b,c\le a \iff \max\{b,c\}\le a \iff p^{\large \max\{b,c\}}\mid p^{\large a}$$

Reassembling the prime components yields the result for the lcm

$$\quad B,C\mid A \iff \color{#c00}{p^{\max\{b,c\}} q^{\max\{b',c'\}}}\ldots\mid A$$

Remark $\ $ Above we employ the universal characterization of lcm, i.e.

$$\begin{align} \ \ \ \ \ B,C\mid A\iff \color{#c00}{{\rm lcm}(B,C)}\mid A\end{align}\qquad\qquad$$