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 ∗?

Question

Philoid · Accepted 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 *.

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 ∗?