Recently, at a math competition, I was given the following question: Determine the smallest number that gives a perfect cube when multiplied by $180$ . I had thirty seconds to solve this question and no calculator.
The answer was $150$ since $150 \cdot 180=27000$ and $\sqrt[3]{27000}=30$.
I was stuck on this question without a calculator. Using a calculator with graphing and table generating capabilities, one could simply put $f(x)=\sqrt[3]{180x}.$ Then, they could scroll through a table until they found that $f(150)=30$, an integer. However, I don't see how this could be done without a calculator. Even further, if one did have a calculator, how it could be done in thirty seconds.
How could this be done feasibly in thirty seconds without a calculator? Does it just require enough number sense to know that $180$ is a factor of $27000$?
$180=2^2\cdot3^2\cdot5$; to make a cube you need to multiply by $2\cdot3\cdot5^2=150$, since you need the exponents to be multiples of $3$.