In a recent question we raised the theorem:
for a given prime $p$ and a given power $m$ the representation of any positive integer $n\in \Bbb N$ in the form: $$ n=(a_u p - b_u) \; p^m$$ is unique with the coefficient pairs (OEIS: A226233 A226236) $$ \left\{ \begin{array}{l l} \langle a_u \rangle=1+\left\lfloor\frac{u-1}{p-1}\right\rfloor=\frac{(p-1)+u-1-((u-1)mod(p-1))}{p-1}\\ \langle b_u \rangle=u-(p-1)\left\lfloor\frac{u-1}{p-1}\right\rfloor=1+((u-1)mod(p-1)) \end{array} \right.$$
while $a_u,b_u,u\in \Bbb N$ and $m\in \Bbb N_0$.
Example: $b_u(p=5)={1,2,3,4}$
The questions are:
1. Is it mathematically correct to state that the sequence $b_u$ represents the ring of the residual classes $(\Bbb Z/p\Bbb Z)^\times$?
2. Is it correct to state that $b_u$ represents the coefficients $\alpha_i$ of corresponding $p$-adic integers according the common general description $\sum_{k=q}^\infty \alpha_i p^k$?
3. How may one interpret $n=(a_u p - b_u) \; p^m$ then through the description $\sum_{k=q}^\infty \alpha_i p^k$?