Finding inverse mod $p^{k}$

Question

We can find the inverse mod $p$ by taking a primitive root and generating the multiplicative group with it. Is there a similar method for finding the inverse of the invertible elements mod $p^{k}$, $k>1$?

Answer 1

If $p \ne 2$, you can use the same method. The inverse in any cyclic group can be found by the same method. The invertible elements mod $p^k$ (for $p \ne 2$) form a cyclic group (under multiplication).

Answer 2

If prime $p \ne 2$, there is a primitive root mod $p^k$.