Congruence Modulo Power of p as Linear Combination of Congruences Modulo p

Theorem
Let $p$ be a prime number.

Let $S = \set {a_1, a_2, \ldots, a_p}$ be a complete residue system modulo $k$.

Then for all integers $n \in \Z$ and non-negative integer $s \in \Z_{>0}$, there exists a congruence of the form:


 * $n \equiv \displaystyle \sum_{j \mathop = 0}^s b_j p^j \pmod {p^{s + 1} }$