Question

Let * be the binary operation on N given by ab=L.C.M. of a and b. Find
(i) 57, 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

The binary operation * on N is 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

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(ii) It is known that: L.C.M of a and b = L.C.M of b and a ∀ a, b ∈ N.
∴a * b = b * a
Thus, the operation * is commutative.

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(iii) For a, b, c ∈ N, we have:
(a * b) * c = (L.C.M of a and b) * c = LCM of a, b, and c a * (b * c) = a * (LCM of b and c) = L.C.M of a, b, and c
∴(a * b) * c = a * (b * c)
Thus, the operation * is associative.

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(iv) It is known that:
L.C.M. of a and 1 = a = L.C.M. 1 and a ∀ a ∈ N
⇒ a * 1 = a = 1 * a ∀ a ∈ N
Thus, 1 is the identity of * in N.

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(v) An element a in N is invertible with respect to the operation * if there exists an
element b in N, such that a * b = e = b * a.
Here, e = 1
This means that:
L.C.M of a and b = 1 = L.C.M of b and a
This case is possible only when a and b are equal to 1.

Thus, 1 is the only invertible element of N with respect to the operation *.