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By what least number 4320 be multiplied to become a perfect cube?
A.10
B.30
C.20
D.50

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Last updated date: 20th Jul 2024
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Answer
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Hint: In this problem, we have to find the least number, which when multiplied to 4320, gives a perfect cube. We can first find the prime factors for the given term 4320, we can then combine them in triplets and we check for the missing terms which should be in the triplets as it should be a perfect cube, we can then multiply the terms which we had combined to make triplets, to get the final answer.

Complete step by step answer:
Here we have to find the least number, which when multiplied to 4320, gives a perfect cube.
We can now find the prime factors of 4320, we get
\[\begin{align}
  & 2|\underline{4320} \\
 & 2|\underline{2160} \\
 & 2|\underline{1080} \\
 & 2|\underline{540} \\
 & 3|\underline{270} \\
 & 3|\underline{90} \\
 & 3|\underline{30} \\
 & 5|\underline{10} \\
 & 2|\underline{2} \\
 & 1|\underline{1} \\
\end{align}\]
We can now write it in order, we get
\[\Rightarrow 4320=2\times 2\times 2\times 2\times 2\times 3\times 3\times 3\times 5\times 1\]
We can now write in terms of power, we get
\[\Rightarrow 4320={{2}^{5}}\times {{3}^{3}}\times {{5}^{1}}\]
We know to get a cubic term, we should have all the terms in triplets.
Here we can see that, we need one 2 and two 5’s, to get the triplet form, i.e. cubic form
\[\Rightarrow 2\times 5\times 5=50\]
We can now check, whether we get perfect cube if we multiply 50 to 4320,
\[\Rightarrow 4320\times 50=216000\]
We can now apply cubic root,
\[\Rightarrow \sqrt[3]{216000}=60\]
The resulting value is a perfect cube, when 50 is multiplied to 4320.

So, the correct answer is “Option D”.

Note: We should always remember that if we have all the terms in triplets, then we will get a perfect cube term. We should also know to find the prime factors for the given number to get the individual term and to find which term can be multiplied to it, to get the final answer.