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)