Question

Find the smallest number which when subtracted from $9761$, the difference is a perfect cube.

Answer 1

The largest perfect cube less than 9761 will differ from 9761 by a smaller amount than any smaller perfect cube will differ from it.

$\Rightarrow$ We take the cube root of 9761.

$\Rightarrow \sqrt[3]{39761}=21.37132445$

$\Rightarrow$ The largest integer less than $21.37132445$ is 21.

So ${21}^3$ or 9261 is the largest perfect cube that's less that 9761.

$\therefore 9761-9261=500$

Hence, 500 is the smallest number that can be subtracted from 9761 so that the difference, 9261, is a perfect cube and the cube root of that difference is 21.

Hence, the correct answer is 500.