I agree that Mathman and I were probably reading the problem incorrectly, because given thatI've just read Dodo's note and the previous replies now make more sense. But I think what he says is certainly correct. It is the case that
[itex](\exists k\in N)b=a^k\implies(\forall n\in N)a^n-1|b^n-1[/itex]
I think we're being asked to prove the converse. The question wouldn't make much sense read the other way.
The converse is likely to be a lot more awkward, which is why I wanted to get an idea of how much more awkward before I started thinking about it.