We know that a * b = LCM of a and b
So 20 * 16 = LCM of 20 and 16 = 80
Commutative:
Consider a and b ∈ N
So LCM of a and b = LCM of b and a
Here, a * b = b * a Ɐ a and b ∈ N
Therefore, * is commutative.
Associative:
(a * b) * c = LCM of a and b * c
Here,
= LCM of {(LCM of a and b) and C} = LCM of {LCM of a, b and c}
We get
= LCM of (a and [LCM of b and c]) = a * (LCM of b and c)
So
= a * (b * c)
We can write it as
a * (b * c) = (a * b) * c Ɐ a, b ∈ c ∈ N.
Therefore, * is associative.