# 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!

You can find here that all such lattices are distributive (that is, lattices of divisors of some integer).
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.

Here you see that a complemented distributive lattice is Boolean.
Thus, if $D_{150}$ is distributive and not Boolean, it is not complemented.
Your identification of the complements seems correct.