* is an operation as m*n = LCM (m, n) where m, n ∈ N. Let m = 2 and b = 3 two natural numbers.
m*n = 2*3
= LCM (2, 3)
= 6∈ N
So, * is a binary operation from N x N → N.
For commutative,
n*m = 3*2
= LCM (3, 2)
= 6∈ N
Since m*n = n*m, hence * is commutative operation.
Again, for associative, let p = 4
m*(n*p) = 2*LCM (3, 4)
= 2*12
= LCM (2, 12)
= 12∈ N
(m*n) *p = LCM (2, 3) *4
= 6*4
= LCM (6, 4)
= 12∈ N
As m*(n*p) = (m*n) *p, hence * an associative operation.
