Question

Define * on N by m * n = 1cm (m, n). Show that * is a binary operation which is commutative as well as associative.

Answer 1

Consider m and n ∈ N where m * n = LCM (m, n) = LCM (n, m) = n * m

* is commutative binary operation.

We know that

(m * n * p = [LCM of (m, n)] * p = [LCM of (m, n) and p] = LCM of (m, n, p)

Similarly

m * (n * p) = m * [LCM of (n, p)] = LCM of [m and LCM of (n, p] = LCM of (m, n, p)

So we get (m * n) * p = m * (n * p)

Therefore, the operation is associative.