Proving (n-1)|(n^k - 1) and the Primality of n^k - 1 when n=2 and k is Prime

[Fairy111]

Homework Statement

Let n and k be integers with n>=2 and k>=2. Prove that (n-1)|(n^k - 1).
Hence prove that if n^k - 1 is prime then n=2 and k is prime.

Homework Equations

The Attempt at a Solution

I think you go about this question by using proof by induction. However I am really not sure how to do this. Any help would be great! Thanks

 

[yyat]

Hint: (n-1)(n^{k-1}+n^{k-2}+\ldots+n+1)=n^k-1

 

[Fairy111]

its supposed to be n^(k) - 1

 

[yyat]

its supposed to be n^(k) - 1

Isn't that the same as the RHS of the formula I wrote?