I'm asked the following question: Prove that $b$ is a primitive root modulo $p \implies$ the smallest positive exponent $e$ such that $b^e \equiv 1 \pmod p$ is $p - 1$.
I know that this could probably be shown easily with Fermat's Little Theorem, but it's posed to the reader before Fermat's Little Theorem is even discussed.
How should I approach this problem? Is there a simple argument for why this is true that doesn't require quite as much ingenuity as the proofs of Fermat's Little Theorem? (i.e., the ones that involve $(p-1)!$ or group theory)