Let * be a binary operation on N given

Question:

Let * be a binary operation on N given by a *b = 1 cm of a and b. Find the value of20 * 16. Is *

(i) commutative,

(ii) associative?

 

Solution:

To find: LCM of 20 and 16

Prime factorizing 20 and 16 we get. 

$20=2^{2} \times 5$

$16=2^{4}$

$\Rightarrow$ LCM of 20 and $16=2^{4} \times 5=80$

(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

$\Rightarrow \mathrm{LCM}[\mathrm{LCM}(\mathrm{a}, \mathrm{b}), \mathrm{c}]=\mathrm{LCM}[\mathrm{a}, \mathrm{LCM}(\mathrm{b}, \mathrm{c})]$

Let the prime factors of a be $p_{1}, p_{2}$

Let the prime factors of $b$ be $p_{2}, p_{3}$

Let the prime factors of $c$ be $p_{3}, p_{4}$

Let the higher factor of $p_{i}$ be $q_{i}$ for $i=1,2,3,4$

$\operatorname{LCM}(a, b)=p_{1}^{q 1} \times p_{2}^{q 2} \times p_{3}^{q 3}$

$\operatorname{LCM}[\operatorname{LCM}(a, b), c]=p_{1}^{q 1} \times p_{2}^{q 2} \times p_{3}^{q 3} \times p_{4}^{q 4}$

$\operatorname{LCM}(b, c)=p_{2}^{q 2} \times p_{3}^{q 3} \times p_{4}^{q 4}$

$\operatorname{LCM}[a, L C M(b, c)]=p_{1}^{q 1} \times p_{2}^{q 2} \times p_{3}^{q 3} \times p_{4}^{q 4}$

$\Rightarrow^{*}$ is associative 

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