question_answer 51)

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

Answer:

Given a * b = H.C.F. of a and        and a and b       = H.C.F. of b and a = b * a              is commutative.       Also (a * b) * c = (H.C.F. of a and b) * c             = a * (H.C.F. of b and c)       = a * (b * c)        is associative.       Let e be the identity element of N.                H.C.F. of a and e =        a divides       But there is no number ?e? which is divisible by every natural number.       Hence identity element does not exist in N.