Prove that if $n$ is a positive integer then $2^{3n}-1$ is divisible by $7$. I encountered a problem in a book that was designed for IMO trainees. The problem had something to do with divisibility. 

Prove that if $n$ is a positive integer then $2^{3n}-1$ is divisible by $7$.

Can somebody give me a hint on this problem. I know that it can be done via the principle of mathematical induction, but I am looking for some other way (that is if there is some other way)

HINT: $2^{3n}=8^n$, and for integers $n\ge2$ there is a well-known factorization of $x^n-y^n$.
