Proving a property with the Euclidean division properties

Question

I am trying to prove that any square AND cube number (that is, any number which is the square and the cube of other two numbers, for example $64=8^2=4^3$, I don't know if this has a proper nomenclature since I'm not native) can be written as either $7k $ or $ 7k+1$

The problem here is that I can only use the Euclidean division property to prove it:

Given two integers a and b, with b ≠ 0, there exist unique integers q and r such that

a = bq + r

and

0 ≤ r < |b|

I also guess I can use the following properties related to the problem, which I managed to prove beforehand:

The square of any number can be written as 3k or 3k+1

The cube of any number can be written as 9k or 9k+1 or 9k+8

Using this, I have tried to analyse the remainders $r=0$ and $r=1$ for $a=7q+r$, then study $a^2$ and $a^3$ with said expression on each case and check if they are divisible for any of their respective possible expressions as square or cube, but I was not able to prove any of them apart from $a^2$ in $r=0$

I am not sure if I should prove it this way, if I am missing something or if there is any other way to do this, help is welcome!

Answer 1

Hint:

A number which is simultaneously square and cube is necessarily a number which is a perfect power of six.

Continuing with your thought process of looking at modulo $7$, what are the possible values of $a^6\pmod{7}$?

$~$

Note that $6=7-1$, that $7$ is prime, and remember Fermat's little theorem