Invertible elements of $\mathbb{Z}/p^n\mathbb{Z}$ Let $p \ge 3$ be a prime and let us consider $U(p^n)$, namely the set of all invertible elements of $\mathbb{Z}/p^n\mathbb{Z}$. Is it true that $1+p \in U(p^n)$? If so, which is its order in $U(p^n)$?
HINT:

$$[k]_{p^n} \in U(p^n) \iff (k,p^n) = 1$$

where $[k]_{p_n}$ is an element of the quotient group with representant $k \in \mathbb{Z}$
The order of $U(p^n)$, $n\geq 1$, is given by Euler's totient function $\phi$ with $\phi(p^n)=p^n-p^{n-1}$.