Question

Is 9720 a perfect cube? If not, find the smallest number by which it should be divided to get a perfect cube.

Answer 1

Hint: To check whether given number 9720 is a perfect cube or not, we will find its factor by finding its least common factor. We check for the triplets formed by the factors of the number. If all the factors form a triplet, then the number can be categorised as a perfect cube.
If not, then we will check for the number which are not forming a triplet. To get a number which is a perfect cube, we will divide the given number by the numbers which are not forming triplets in the factors.


Complete step by step solution
To check whether the given number 9720 is a perfect cube or not, we find will find its factor by finding its least common factor which can be expressed as:
$\begin{array}{l}
9720 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 5\\
9720 = \underline {2 \times 2 \times 2} \times \underline {3 \times 3 \times 3} \times 3 \times 3 \times 5
\end{array}$

Since, we can see that there is no triplet for 3 and 5 in the above expression, hence the number 9720 is not a perfect cube.

 We can see that $3 \times 3 \times 5$ or 45 is the number because 9720 is not a perfect cube.

Hence, we will divide our number with $45\left( {3 \times 3 \times 5} \right)$ we can get the number which is a perfect cube. Since we will be left with two triplets. This can be expressed as:
$\begin{array}{l}
\dfrac{{9720}}{{45}} = \dfrac{{2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 5}}{{3 \times 3 \times 5}}\\
216 = \underline {2 \times 2 \times 2} \times \underline {3 \times 3 \times 3} \\
216 = {\left( {2 \times 3} \right)^3}\\
216 = {6^3}
\end{array}$

As 216 is the perfect cube, therefore, 45 is the smallest number by which the number 9720 should be divided to get a perfect cube.

Note: To check whether a number is a perfect square or not, we have two methods, one is through a long division method and other is by finding its factors. But to check whether a number is a perfect cube or not, we will only check for triplets in the factors.