How to prove $k^n \equiv 1 \pmod {k-1}$ (by induction)? How to prove $k^n \equiv 1 \pmod {k-1}$ (by induction)?

Well, I'll leave the case of $n=1$ to you.
So, for a fixed $k$, suppose that $k^n\equiv 1 \mod(k-1)$ for some $n\in \mathbb{N}$.
We want to show that $k^{n+1} \equiv 1 \mod(k-1)$.  Well, $k^{n+1}=k^n k$, and we know that $k^n\equiv 1 \mod(k-1)$ (since this is the induction hypothesis).  So, what is $k^{n+1}$ congruent to mod $k-1$?
