The book that I'm reading (Elementary Number Theory by Underwood Dudley) gives a Theorem: If $p$ and $q$ are odd primes and $q|a^p-1$, then either $q|a-1$ or $q=2kp+1$, for some integer $k$.
Then it gives a corollary; Any divisor of $2^p-1$ is of the form $2kp+1$.
Then it gives an exercise; Using the corollary, what is the smallest prime divisor of $2^{19}-1$?
So I go through $k=1,2,3,4,5,6,7,...$ and I find nothing....So I check the back of the book and it says that $191$ is the smallest prime divisor ($191=2\cdot5\cdot19+1$) since when $k=1..4$ the numbers are composite. Yet $191\not|2^{19}-1$ since, after checking, $2^{19}-1$ is itself a prime number; $524287$.
So I ask, what have I done wrong to screw this up? I can not for the life of me understand where my calculation errs... I've checked a couple of times and $2^{19}-1$ is a Mersenne Prime...