Clarification on what this relation means for Hasse diagram

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.

• It is in fact a vague relation. Does it mean distinct prime factors or total prime factors? I would bet on the second one, though. – u1571372 Nov 11 '16 at 1:07
• @u1571372 I'm not too sure either but I also don't know if I'm restricted to only the numbers in the set – Sam Nov 13 '16 at 16:42

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.

• Hmm this does seem to make sense but I have to make a diagram so it guessing the ones that only have 1 prime number will be placed at the top – Sam Nov 12 '16 at 3:55
• No, do you understand what is the covering relation? The poset has four "levels" and in each level each number is less than every element in the level above. The first level has $2$ and $7$; the second $21$, $33$ and $201$; the third $30$; and the fourth has $330$ and $390$. – amrsa Nov 12 '16 at 9:19
• But then 67 isn't a prime factor of 210? Isn't it supposed to be 2,3,5,7 – Sam Nov 13 '16 at 15:40
• I don't understand. There is no $210$; did you mean $201$? And what do you mean with $2,3,5,7$? The product of these is $350$ which doesn't belong to the set. Also, keep in mind that the order is not about divisibility, but about number of prime factors. – amrsa Nov 14 '16 at 10:31
• Oops. I meant 210 that was a typo at the beginning. – Sam Nov 14 '16 at 20:39