Recall the topic “Ordered sets and Lattices” that the set $D_m$ of divisors of $m$ is a bounded, distributive lattice with $$a+b = a\lor b =\operatorname{lcm}(a, b)$$ $$ab = a\land b =\gcd(a, b)$$
(a) Show that $D_m$ is a Boolean algebra if $m$ is square free, i.e., if $m$ is a product of distinct primes.
(b) Find the atoms of $D_m$.
Since the divisors of m area bounded, distributive lattice, all that is left to show they are a Boolean algebra is to create complements.
As m is square free, an element a is a product of distinct primes.
Show a', the complement of a, is the product all the prime divisors of m that do not divide a.
Clearly the atoms are the primes.
Exercise. If m is not square free, show that some elements do not have a complement.