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Let * be a binary operation on N given by a * b = 1 cm of a and b. Find the value of 20 * 16. Is * (i) commutative, (ii) associative?

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We know that a * b = LCM of a and b

So 20 * 16 = LCM of 20 and 16 = 80

Commutative:

Consider a and b ∈ N

So LCM of a and b = LCM of b and a

Here, a * b = b * a Ɐ a and b ∈ N

Therefore, * is commutative.

Associative:

(a * b) * c = LCM of a and b * c

Here,

= LCM of {(LCM of a and b) and C} = LCM of {LCM of a, b and c}

We get

= LCM of (a and [LCM of b and c]) = a * (LCM of b and c)

So

= a * (b * c)

We can write it as

a * (b * c) = (a * b) * c Ɐ a, b ∈ c ∈ N.

Therefore, * is associative.

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