B and D both are the answers. (I think, Verification required.)
a.) | is an equivalance relation. False
3 | 6 but not the other way around. so not symmetric
b.) Every subset of N has an upper bound under |. True
Every finite subset A does, it is lcm(A) .
Also even infinite subsets of $\mathbb{N}$ have least upper bound if we count 0 as natural number, (surprised !!) because everything divides zero.
Source: https://en.wikipedia.org/wiki/Complete_lattice. see examples.
c.) | is a total order. False
3 and 5 are not comparable.
Defination of total order: A poset $(S, \preceq)$ is total order if $\forall{x,y \in S}$ either $x \preceq y$ or $y \preceq x$
d.) (N, |) is a complete lattice. True
Option b explains the reason for upper bound. For finite subset we have gcd as lower bound, but for infinite subets we always have 1, if no other exists. :-)
e.) (N,∣) is a lattice but not a complete lattice. False.
Now it's obvious, isnt' it. :-)
EDIT: I looked in an answer key. The answer as per the key is E. I guess they are not counting 0 as natural number. Which implies there is no upper bound for infinite subset, which makes both B and D false.