Show that if $x$ is a primitive root modulo $p^r$, then $x$ is a primitive root modulo $p$

Question

I'm trying to solve the following problem:

Let $p\geq 3$ be a prime number, let $r \in \Bbb N$, and let $x$ be a primitive root modulo $p^r$. Show that $x$ is a primitive root modulo $p$.

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I'm pretty much out of ideas. I tried to use the following claim: $a$ is a primitive root modulo $n$ if and only if for every prime $q$ such that $q$ divides $\varphi(n)$ we have: $$a^{\frac{\varphi(n)}{q}}\not\equiv 1\pmod n$$

But got nothing.

Answer 1

Hint: If $x^{\frac{p-1}{2}} \equiv 1 \bmod p$, then $x^{\frac{p-1}{2}(p^{r-1})} \equiv 1 \bmod p^r$ by induction on $r$.