Q. 64.0( 12 Votes )

Let be the 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 :

(i) It is given that the binary operation on N given by a b = L.C.M. of a and b.

Then, 5 * 7 = LCM of 5 and 7 = 35


20 * 16 = LCM of 20 and 16 = 80.


(ii) It is given that the binary operation on N given by a b = L.C.M. of a and b.


We know that LCM of a and b = LCM of b and a, a,b ϵ N.


a * b = b * a


Therefore, the operation * is commutative.


(iii) It is given that the binary operation on N given by a b = L.C.M. of a and b.


For a, b, c ϵ N


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


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


(a * b)* c = a * (b * c)


Therefore, the operation * is associative.


(iv) It is given that the binary operation on N given by a b = L.C.M. of a and b.


We know that LCM of a and 1 = a = LCM of 1 and 1, a ϵ N


a * 1 = a = 1 * a, a ϵ N


Therefore, 1 is the identity of * in N.


(v) It is given that the binary operation on N given by a b = L.C.M. of a and b.


An element a in N is invertible w.r.t. the operation * if there exists an element b in N,


Such that a * b = e =b * a


Now, if e = 1


LCM of a and b = 1= LCM of b and a


This is only possible when a = b = 1


Therefore, 1 is the only invertible element of N w.r.t. the operation *.


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