Clarification on what this relation means for Hasse diagram

Question

My question reads: the relation "has fewer prime factors" on the set {2,7,21,30,33,210,330,390}. I'm confused how to approach it to draw the Hasse Diagram mostly due to what's in quotes.

Answer 1

The way I see it (and it seems straightforward), you just have to decompose the numbers in prime factors and order the numbers according to the number of prime factors.

Now, this gives

\begin{align} \textbf{number} &&\textbf{prime factors} &&\textbf{# prime factors}\\ 2 &&2 &&1\\ 7 &&7 &&1\\ 21 &&3,7 &&2\\ 30 &&2,3,5 &&3\\ 33 &&3,11 &&2\\ 201 &&3,67 &&2\\ 330 &&2,3,5,11 &&4\\ 390 &&2,3,5,13 &&4 \end{align}

So the covering relation of this poset is given by $$(2,21), (2,33), (2,201), (7,21), (7,33), (7,201), (21,30), (33,30), (201,30), (30,330), (30, 390).$$

Note. I previously had two different approaches, but erased one of them (the one which occurred me first) because the second one (present here) seems to me to correspond to the right interpretation of the question.