What is the smallest number x such that $180\times x$ is a perfect cube?

Question

What is the smallest number x such that $180\times x$ is a perfect cube?

Answer 1

$180=2^2 3^2 5$ so the smallest $x$ is $2 \cdot 3\cdot 5^2=150$

Answer 2

Hint

We have

$$180=2^2\times 3^2\times 5$$ so we choose $x=2\times 3\times 5^2$. Why?

Answer 3

$$180 = 2^2 \cdot 3^2 \cdot 5,$$ So multiplying by $$x = 2\cdot 3\cdot 5^2 = 150$$ gives us $$180 x = \underbrace{(2^2\cdot 3^2 \cdot 5)}_{\large =\,180} \cdot \underbrace{(2 \cdot 3\cdot 5^2)}_{\large =\,x} = 2^3\cdot 3^3 \cdot 5^3 = (2\cdot 3\cdot 5)^3 = (30)^3 = 27000$$

as desired.