The binary operation * on N defined as:

a * b = H.C.F. of a and b, a, b N.

Therefore,

b * a = H.C.F. of b and a = H.C.F

Hence, * is commutative on N.

Let a, b, c N.

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

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

For example, take the numbers 2, 4, 7 N.

Here, if the operation is applied to the above numbers as follows:

(2 * 4) * 7 = 1 * 7 = 1 (1 is the HCF of 2, 4 and HCF of 1, 7 is 1). Also,

2 * (4 * 7) = 2 * 1 = 1 (Similar as the above reason)

Therefore, * is associative.

Now, Identity Element:

Given a binary operation *: N X N → N, an element e ∈A, if it exists, is called an identity of the operation * , if a * e = a = e * a ∀ a ∈A.

But there does not exist any value of e ∈N such that a * e = a = e * a

Therefore, this operation does not have any identity.