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?

To find: LCM of 20 and 16

Prime factorizing 20 and 16 we get.

20 = 2² × 5

16 = 2⁴

⇒ LCM of 20 and 16 = 2⁴ × 5 = 80

(i) To find LCM highest power of each prime factor has been taken from both the numbers and multiplied.

So it is irrelevant in which order the number are taken as their prime factors will remain the same.

So LCM(a,b) = LCM(b,a)

So * is commutative

(ii) Let us assume that * is associative

LCM[LCM(a,b),c] = LCM[a,LCM(b,c)]

Let the prime factors of a be p₁,p₂

Let the prime factors of b be p₂,p₃

Let the prime factors of c be p₃,p₄

Let the higher factor of p_i be q_i for i = 1,2,3,4

LCM (a,b) = p₁^q1p₂^q2p₃^q3

LCM[LCM(a,b),c] = p₁^q1p₂^q2p₃^q3p₄^q4

LCM (b,c) = p₂^q2p₃^q3p₄^q4

LCM[a,LCM(b,c)] = p₁^q1p₂^q2p₃^q3p₄^q4

* is associative