Question

What is the smallest number by which $18522$ must be divided so that the quotient is a perfect cube?

Answer 1

On Prime factorization of $18522$, we get,

$18522=2\times 3\times 3\times 3\times 7\times 7\times 7$

$=2^1\times 3^3\times 7^3$.

We know, a perfect cube should have each prime factor in power of $3$.

Here, number of $2$'s is in power of $1$, number of $3$'s is in power of $3$ and number of $7$'s is in power of $3$.

So, we need to remove $2$ from the factors to make $18522$, a perfect cube.

Hence, the smallest number by which $18522$ must be divided to obtain a perfect cube is $2$.