Let* be the binary operation on N given by a * b = L.C.M. of a and Fin

1.	Let* be the binary operation on N given by a * b = L.C.M. of a and 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) 5 7 = L.C.M. of 5, 7 = 35 20 16 = L.C.M. of 20, 16 = 80. (ii) a * b = L.C.M. of (a, b) = L.C.M. (b,a) = b * a hence * is commutative. (iii) (a * b) * c = L.C.M. of a, b, c a* (bc) = L.C.M. of a, b, c hence is associative. (iv) Let ‘e’ be the identity element, than a * e = e * a = a ∀ a ∈ N L.C.M. (a, e) = L.C.M. (e, a) = a ⇒ e divides a ∀a ∈N ⇒ e = 1 ∴ 1 is the identity element. (v) Let a ∈ N be invertible. ∴ be N such that a * b = b * a = L.C.M. (a, b) = 1 ⇒ a = 1, b = 1 ∴ only invertible element in N is 1

Let* be the binary operation on N given by a * b = L.C.M. of a and 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) 5 *7 = L.C.M. of 5, 7 = 35

20 *16 = L.C.M. of 20, 16 = 80.

(ii) a * b = L.C.M. of (a, b) = L.C.M. (b,a) = b * a

hence * is commutative.

(iii) (a * b) * c = L.C.M. of a, b, c

a* (b*c) = L.C.M. of a, b, c

hence * is associative.

(iv) Let ‘e’ be the identity element, than

a * e = e * a = a ∀ a ∈ N

L.C.M. (a, e) = L.C.M. (e, a) = a

⇒ e divides a ∀a ∈N ⇒ e = 1

∴ 1 is the identity element.

(v) Let a ∈ N be invertible.

∴ be N such that a * b = b * a = L.C.M. (a, b) = 1

⇒ a = 1, b = 1

∴ only invertible element in N is 1