question_answer 49)

Let * be binary operation on N given by a * b = L.C.M. of a and b. Find (i) 5 * 7, 20* 16 (ii) Is * commutative ? (iii) Is * associative ? (iv) Find the identity of * in N. (v) Which elements of N are invertible for the operation *?  

Answer:

Given a * b = L.C.M. of a and b (i)     5 * 7 = L.C.M. of 5 and 7 = 35       20 * 16 = L.C.M. of 20 and 16 = 80 (ii)   a * b = L.C.M. of a and b       = L.C.M. of b and a       = b * a             Yes, * is commutative. (iii)  a * (b * c) = a* (L.C.M. of b and c)       = L.C.M. of a, b and c       = (L.C.M. of a and b) * c       = (a * b) * c        = (a * b) * c       Yes, * is associative. (iv)  Let e be the identity element of in N              of a and e =                    is the identify element. (v)   Let ?b? be the invertible element in N        = 1 (identity element)       L.C.M. of a and b ? 1       a = b = 1        is an invertible element in N.