Question

By what smallest number should 216 be divided so that the quotient is a perfect square? Also, find the square root of the quotient.

Answer 1

Hint: To solve this question, we will first find the factor of 216 by taking the LCM of 216. The LCM is the least common multiple of the number. Then after going for the LCM of 216, we will observe which are the numbers such that 216 is not able to become a perfect square. And finally, we will divide it by that number so as to get a number as a perfect square.

Complete step by step answer:
We are given the number as 216. First of all, let us determine the factors of 216 by computing its LCM. The LCM of 216 is given as
\[\begin{align}
  & 2\left| \!{\underline {\,
  216 \,}} \right. \\
 & 2\left| \!{\underline {\,
  108 \,}} \right. \\
 & 2\left| \!{\underline {\,
  54 \,}} \right. \\
 & 3\left| \!{\underline {\,
  24 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }1 \\
\end{align}\]
Therefore, the factors of 216 are given as
\[216=2\times 2\times 2\times 3\times 3\times 3\]
Let us form pairs of factors. Pairing the factors of 216, we get,
\[216=\underline{2\times 2}\times \underline{3\times 3}\times 2\times 3\]
So, we see that a term \[2\times 3\] is left unpaired. This term \[2\times 3\] is restricting 216 to become a perfect square. Therefore, the number should be divided by \[2\times 3\] to become a perfect square.
\[2\times 3=6\]
Therefore 216 by 6, we get,
\[\Rightarrow \dfrac{216}{6}=36\]
And then factors of 36 are
\[36=2\times 2\times 3\times 3\]
All are paired here, so 36 is a perfect square. Hence, we should divide 216 by 6, so as to get the quotient as a perfect square. The square root of the quotient is
\[\sqrt{36}=\sqrt{6\times 6}=\sqrt{{{6}^{2}}}=6\]
Hence, we have obtained the answer.

Note: The square root of the quotient 36 can also be determined using the long division method of the square root. It is given as below.
\[6\overset{6}{\overline{\left){\begin{align}
  & \overline{36} \\
 & \underline{36} \\
 & 0 \\
\end{align}}\right.}}\]
Hence the answer is \[\sqrt{36}=\pm 6.\]