Question

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 1

(i) Given, a*b = L.C.M of a and b
5*7 = L.C.M of 5 and 7 = 35
20*16 = L.C.M of 20 and 16 = 80

(ii) we can see that ,
a*b = L.C.M of a and b
b*a = L.C.M of b and a
we know, L.C.M of a and b = L.CM of b and a
so, a*b = b*a
therefore, * is commutative.

(iii) Let's take 1,2,3 ∈ ℕ
(1*2)*3 = {L.C.M of 1 and 2}*3
= 2*3 = L.C.M of {2 and 3 } = 6
1*(2*3) = 1*{L.C.M of 2 and 3 }
= 1*6 = L.C.M of 1 and 6 = 6
e.g., (1*2)*3 = 1*(2*3)
therefore, * 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 *.