Question

Find the smallest number which divides $88209$ so that the quotient is a perfect cube.

Answer 1

To find the smallest number by which 88209 must be divided so that the quotient is a perfect cube, we have to find the prime factors of 88209.

$88209=3\times 3\times 3\times 3\times 3\times 3\times 11\times 11$

Prime factors of 88209 are $3,3,3,3,3,3,11,11$.

Out of the prime factors of 88209, 11 cannot be considered in its perfect cube as it have only two factors of 11.

So, $11\times 11$ is the number by which 88209 must be divided to make the quotient a perfect cube.

$\Rightarrow \frac{88209}{11\times 11}=729$

$\sqrt[3]{3729}=9$

Hence, the smallest number is 121, which when divides 88209, the quotient is 729 which is a perfect cube.

Hence, the correct answer is $121.$