Find the smallest number by which $128$ must be divided to obtain a perfect cube?

The prime factorization gives:

$128=2\times 2\times 2\times 2\times 2\times 2\times 2$

Now, if we combine the elements into triples of equal factors, we can see that

$128=(2\times 2\times 2)\times (2\times 2\times 2)\times 2$

Here, $2$ cannot be grouped into triples of equal factors.

Therefore, we will divide $128$ by $2$ to get a perfect cube.