Question

Find the smallest number by which the following number must be divided to obtain a perfect cube: $192$

Answer 1

Calculate the prime factorization of $192$.

$192=2\times 2\times 2\times 2\times 2\times 2\times 3$

To make it a perfect cube, it should be divided by $3$ so that the factors can be grouped into triplets

$\begin{array}{rcl}\frac{192}{3}& =& \frac{2\times 2\times 2\times 2\times 2\times 2\times 3}{3}\\ & =& 2\times 2\times 2\times 2\times 2\times 2\\ & =& 4^3\\ & =& 64\end{array}$

which is a perfect cube of $4$.

Hence, the smallest number by which $192$ must be divided to obtain a perfect cube is $3$