Let $p=2q+1$ and $q$ be prime numbers, with $q\equiv 3\:(4)$. Show that $2^q\equiv 1\:(p)$.
So far I've tried to write $q$ as an expression of $p$ and I am assuming that I have to use little Fermat somehow, since $p$ is prime. But I can't figure out how to combine those two and I fail to see an alternative. I also don't see how to use the fact that $q\equiv 3\:(4)$.
I've also tried rewriting the equation to $2^q-1\equiv 0\:(p)$ from which I need to show that $p=2q+1|2^q-1$. But I also got stuck there.
Any help is appreciated.