Question

A number $x$ is equal to $7\cdot24\cdot48$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube?

Answer 1

Take the prime factorization of $x$.

$x = 7\times24\times48 = 7\times(2^3\times3)\times(3\times2^4) = 2^7\times3^2\times7$

If $xy$ is a perfect cube, then the smallest $y$ that makes this happen is $y=2^2\times3\times7^2 = \boxed{588}$. We "complete" the cube by introducing just enough factors to get each prime power to be a multiple of 3.