prove that $(1+pt)^{p^{r-1}} \equiv 1 \pmod {p^r}$ I need to prove that $(1+pt)^{p^{r-1}} \equiv 1 \pmod {p^r}$
the original question is this: 
Prove that , any primitive root $r$ of $p^n$ is also a primitive root of $p$
and I'm following the second answer there. 
I'm trying to use the binomial theorem and having hard time.. 
any help will be appriciated

Here's a good lemma to try to prove. If $s\geq 1$ then:
$$\left(1+kp^s\right)^p\equiv 1\pmod {p^{s+1}}$$
This will let you prove the result above by induction.
