Question

find the smallest number by which 540 must be divided so that it becomes perfect square​

Answer 1

Required Answer:-

Step 1 :- Calculate the prime factors of 540 by using prime factorisation method.

$\begin{array}{c | c} \sf{ \sf \underline{2}}& \sf\underline{540} \\\sf{ \underline{2}} & \sf \underline{270} \\\sf{ \underline{5}}& \sf \underline{135} \\\sf{ \underline{3}} & \sf \underline{27} \sf \\\sf{ \underline{3}}& \sf \underline{9}& \\ \sf{ \underline{3}}& \sf{ \underline{3}} \\ & \sf1\end{array}$

By prime factorization, we got the prime factors as,

→ 540 = 2 × 2 × 5 × 3 × 3 × 3

→ 540 = 2² × 3² × 5 × 3

Here, we got the pairs of 2 and 3 but not for 3 and 5.

Step 2 :- We will multiply 2 and 3 since they have pairs and then after we'll keep 5 × 3 under the root of 2 × 3 since they don't have any pairs.

$\dashrightarrow \: \: \sf \: 2 \times 3 \sqrt{5 \times 3}$

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$\dashrightarrow \: \: \sf \underline{ \boxed{ \tt{ 6 \sqrt{15} }}}$

Conclusion :-

So, we will take 15 which would be multipled by 540 to get a perfect square.

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$\dashrightarrow \: \: \large{\underline{ \boxed{ \tt{ \pmb{ \purple{15}}}}}}$⠀

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Therefore, 15 must be divided so that it becomes a perfect square.

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Knowledge BoosteR :

• In such types of questions in which division and multiplication operation is being performed, there we have to use prime factorisation method and follow same procedures for calculating the answer. The results for both operations will be same.