Let ∗ be the binary operation on N defined by a ∗ b = H.C.F. of a and b. Is ∗ commutative? Is ∗ associative? Does there exist identity for this binary operation on N?

It is given that the binary operation on N defined by a ∗ b = H.C.F. of a and b.

We know that HCF of a and b = HCF of b and a, a, b ϵ N.

⇒ a * b = b * a

⇒ The operation * is commutative.

For a, b c ϵ N, we get,

(a * b) * c = (HCF of a and b) * c = HCF of a, b and c

a * (b * c) = a * (HCF of b and c) = HCF of a, b and c

⇒ (a * b) * c = a * (b * c)

⇒ The operation * is associative.

Now, an element e ϵ N will be the identity for the operation.

Now, if a * e = a = e * a, a ϵ N.

But, this is not true for any a ϵ N.

Therefore, the operation * does not have any identity in N.