Question

By which smallest number must the following numbers be divided so that the quotient is a perfect cube?
$\left(i\right)675$

Answer 1

Step: Find the smallest number​

Resolving $675$ into prime factors:

$$\begin{array}{cc}3& 675\\ 3& 225\\ 3& 75\\ 5& 25\\ 5& 5\\ & 1\end{array}$$

$\Rightarrow 675=\{3\times 3\times 3\}\times 5\times 5$

Clearly, on grouping the prime factors in triplets of equal factors, we observe that the factor $5$ is not forming a triplet.

Thus, we need to divide $675$ by $5\times 5=25$ to make the quotient a perfect cube.

Hence, $25$ is the smallest required number.