Question

Is $2880$ a perfect cube? If not, by which smallest natural number should $2880$ be divided so that the quotient is a perfect cube?

Answer 1

$2880=3\times 3\times 4\times 4\times 4\times 5$

The primes $3$ and $5$ do not appear in groups of three. So, $2880$ is not a perfect cube.

In the factorization of $2880$ the prime $3$ appears only two times and the prime $5$ appears once.

So, if we divide $2880$ by $3\times 5\times 5=75,$ then the prime factorisation of the quotient will not contain $3$ and $5$.

Hence, the smallest natural number by which $2880$ should be divided to make it a perfect cube is $75.$