# Lattice divisors of 150, ordered by divisibility. Draw Hasse diagram, get complements, check if distributive, check if Boolean.

I have the following exercise:

Having the lattice $$D_{150}$$ of the divisors of 150, ordered by divisibility
1) Draw Hasse Diagram;
2) Find all complements;
3) Check if it's a distributive lattice;
4) Check if it's a Boolean lattice.

My development:

The divisors are $$D_{150} = \left \{1,2,3,5,6,10,15,25,30,50,75,150 \right \}$$

point 1)
I have tried to draw the Hasse diagram in this way:

I have also realized that I can draw two cube structures like the following, and I don't know if this fact can help to resolve the exercise:

point 2)
assuming that I have done right the diagram, I have found some complements, but, not every elements have a complement:

$$\begin{array}{c|c} \, & 1 & 2 & 3 & 5 & 6 & 10 & 15 & 25 & 30 & 50 & 75 & 150 \\ \hline 1 & - & - & - & - & - & - & - & - & - & - & - & * \\ \hline 2 & - & - & - & - & - & - & - & - & - & - & * & - \\ \hline 3 & - & - & - & - & - & - & - & - & - & * & - & - \\ \hline 5 & - & - & - & - & - & - & - & - & - & - & - & - \\ \hline 6 & - & - & - & - & - & - & - & * & - & - & - & - \\ \hline 10 & - & - & - & - & - & - & - & - & - & - & - & - \\ \hline 15 & - & - & - & - & - & - & - & - & - & - & - & - \\ \hline 25 & - & - & - & - & * & - & - & - & - & - & - & - \\ \hline 30 & - & - & - & - & - & - & - & - & - & - & - & - \\ \hline 50 & - & - & * & - & - & - & - & - & - & - & - & - \\ \hline 75 & - & * & - & - & - & - & - & - & - & - & - & - \\ \hline 150 & * & - & - & - & - & - & - & - & - & - & - & - \\ \end{array}$$

point 3)
I think it is distributive because I can't find any sublattice like the following nondistributive twos:

point 4)
based on what I have found, since not every element has a complement, then it isn't a Boolean lattice. Or does exists any isomorphism?

please, can you tell what do you think? Many thanks!

Moreover, that they are Boolean iff the number is square-free. In this case, $25|150$, and $25=5^2$, so $150$ is not square-free and $D_{150}$ is not Boolean.
Thus, if $D_{150}$ is distributive and not Boolean, it is not complemented.