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 $3$.

  • $\begingroup$ AH! Should've thought of that... $\endgroup$ – zz20s Jan 11 '16 at 1:24
  • $\begingroup$ @zz20s: I know that feeling well! $\endgroup$ – Brian M. Scott Jan 11 '16 at 2:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.