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 for all a, b ∈ N.