Let ‘*’ be a binary operation on N defined by a*b = L.C.M(a,b) for all a, b ∈N. Find (i) 2*4, 3*5, 1*6 (ii)Check the commutativity and associativity of ‘*’ on N.

LCM Binary Operation Properties

Given Binary Operation

\( a * b = \mathrm{LCM}(a, b) \)

\( 2 * 4 = \mathrm{LCM}(2,4) = 4 \)

\( 3 * 5 = \mathrm{LCM}(3,5) = 15 \)

\( 1 * 6 = \mathrm{LCM}(1,6) = 6 \)

Since:

\( \mathrm{LCM}(a,b) = \mathrm{LCM}(b,a) \)

✔ Operation is commutative

Check:

\( (a*b)*c = \mathrm{LCM}(\mathrm{LCM}(a,b),c) \)

\( a*(b*c) = \mathrm{LCM}(a,\mathrm{LCM}(b,c)) \)

Since LCM satisfies associativity:

✔ Operation is associative

✔ The operation is both commutative and associative on \( \mathbb{N} \).

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