Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / C / C2 / i / Solution

A primitive root modulo $p$ is a generator of the cyclic multiplicative group $(\mathbb{Z}/p\mathbb{Z}{)}^{\times }$, so it has order $p-1$. If $g$ is one generator, the generators are exactly $g^k$ with $\gcd (k,p-1)=1$. Hence there are $\boxed{\varphi (p-1)}$. For $p=2$ the group is trivial and its unique element is its generator, consistently with $\varphi (1)=1$.