Question

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

(i) Find 5 * 7 and 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

Binary operation * defined as 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
b * a = L.C.M. of b and a
= a * b = b * a, since L.C.M. of a, b and b, a are equal.
Binary operation * is commutative.

(iii) a * (b * c) = L.C.M. of a, b, c
and (a * b) * c = L.C.M. of a, b, c
⇒ a * (b * c) = (a * b) * c
⇒ Given binary operation * is associative.

(iv) Identity of * in N is 1 because
1 * a = a * 1 = a = L.C.M. of 1 and a.

(v) Let * : N x N → N defined as a * b = L.C.M. of (a, b)
For a = 1, b = 1, a * b = 1 = b * a. Otherwise a * b * 1.
∴ Binary operation * is not invertible.
⇒ 1 is invertible for operation *.