Is set D together with the following operations a Boolean algebra?

Question

D = {1, 2, 3, 4, 6, 8, 12, 24}

The question is "Is set D together with the following operations a Boolean algebra?"

The operations are as follows.

1. x + y = lcm(x, y)
2. x . y = gcd(x, y)
3. x' = 24/x

My approach is to prove that set D satisfies the following laws in order to show that it forms a Boolean algebra. The laws are, commutative, distributive, associative, and complement. I have proven that D follows commutative, distributive, and associative laws.

But I am confused about proving D obeys the complement law. To prove the complement law, We need to prove, x + x' = 1 and x.x' = 0 isn't it? Can anyone help to prove D follows the complement law?

Any help would be appreciated.

Thanks in advance!