# How do you list all integers less than 20,000 that are both perfect squares and perfect cubes?

Jul 1, 2015

If $n$ is a perfect square and a perfect cube then $n = {k}^{6}$ for some $k$, giving:

${0}^{6} = 0$
${1}^{6} = 1$
${2}^{6} = 64$
${3}^{6} = 729$
${4}^{6} = 4096$
${5}^{6} = 15625$

#### Explanation:

If an integer is both a perfect square and a perfect cube, then it will be of the form ${k}^{6}$ for some $k \in \mathbb{Z}$. You will find that ${5}^{6} < 20000 < {6}^{6}$, so the only possible integers are ${0}^{6}$, ${1}^{6}$, ${2}^{6}$, ${3}^{6}$, ${4}^{6}$ and ${5}^{6}$. (${6}^{6} = 46656$ is too large).