Answer :

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.


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