Question

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?

Answer 1

The binary operation * on N is defined as:
a * b = H.C.F. of a and b
It is known that:
H.C.F. of a and b = H.C.F. of b and a ∀ a, b ∈ N.
∴ a * b = b * a
Thus, the operation * is commutative.
For a, b, c ∈ N , we have: (a * b)* c = (H.C.F. of a and b) * c = H.C.F. of a, b, and c a *(b * c)= a *(H.C.F. of b and c) = H.C.F. of a, b, and c
∴(a * b) * c = a * (b * c)

Thus, the operation * is associative.
Now, an element e ∈ N will be the identity for the operation * if a * e = a = e* a a ∈ N.
But this relation is not true for any a ∈ N.
Thus, the operation * does not have any identity in N.