I am trying to solve this question, however I don´t seem to have the correct expression of the inverse to solve the remaining part:
QUESTION: Find the inverse of 4 in $\mathbb{Z}5$. Use your answer to find an $m \in \mathbb{Z}_{>0}$ such that $4m$ is congruent to 1 mod ($5^4$).
My attempt: I need to find a $k$ such that $4k=1$. Note that $4=3+1$, then using the geometric series for $1/(1-x)$, we have taht $1/4 = 1-3 + 3^2 - 3^3 ...$ which is a 5-adic convergent series.
Now: I Don´t seem to understand how this procedure will help me in the solution of the second part. or weather this is the $m$ that I needed to find.
Anything helps. Thank you in advance!
Hint: $$ \frac{1}{4}= 4+ 3\cdot\frac{5}{1-5}$$