It satisfies all three properties:
Associative : (A+B)+C = A+(B+C)
Let's take A = 2, B =3 , C =5. These all are from set B.
Here + means LCM and * means GCD. So,
LCM(LCM(2,3),5) = 30
Same as LCM(2,LCM(3,5)) i.e. 30. So plus operation is Associative.
GCD(GCD(2,3),5) = GCD(2,GCD(3,5)) = 1 . So Mul operation is Associative.
Commutative : A + B = B + A
LCM(2,3) = LCM(3,2) = 6 and GCD(2,3) = GCD(3,2) =1 So commutative too.
Now lets talk about Distributive,
we need to find A*(B+C) = A*B + A*C
GCD(2,LCM(3,5)) = LCM(GCD(2,3),GCD(2,5)) = 1 So Set B is distributive also.
Correct Me If I am wrong.