Condition inverse $p$-adic number

Question

Take $p$ prime, $n \in \mathbb{Z}_{>0}$ and $x \in \mathbb{Z}_p$.

Suppose that $p$ isn't a divisor of $$x = (x_j + p^j \mathbb{Z})_{j \in \mathbb{Z}_{>0}},$$ then one can prove that the first term $x_1 \neq 0$.

My question is, how does it follow that $x_j \in (\mathbb{Z}/p^j\mathbb{Z})^{\times}$, for any $j \in \mathbb{Z}_{>0}$, without using that it's an integer domain?

Thanks in advance!

Answer 1

If $p$ doesn't divide $x_j$, then $(x_j,p^j)=1$.