Proving De Morgan's Laws for GCD and LCM: A Natural Number Proof

[aufbau86]

Let n be a natural number, and let S be the set of all natural numbers that
divide n.

For a, ,b in S, let gcd(a, b) = a /\ b and lcm(a, b) = a \/ b. For each x
in S, let x' denote n/x. Do de Morgan's laws hold for this system?

(gcd = greatest common divisor, lcm = lowest common multiple)

This is what I have so far for wanting to show (a /\ b)' = a' \/ b'

lcm(n/a, n/b) = a' \/ b'

d = gcd(a, b) = a /\ b.

d' = n/d

So somehow I need to show that n/gcd(a, b) = lcm(n/a, n/b)

From here I've only hit dead-ends.

 

[Nino MMP]

I'm not sure how to proceed in order to show that de Morgan's laws hold for this system. Any help would be greatly appreciated. Thanks!