CR-like-Theorem for powers of same prime

Question

CRT asks the numbers in denominator to be coprime, is there are theorem/property when taking modulo by powers of a prime? It's easy if the power is small but it gets tougher as the number and/or powers grows.

Or is there a way to quickly find modulo , with the help of some related property? For example, $675453 \equiv 3 \pmod 5$ and $675453 \equiv 3 \pmod {25}$ but $675453 \equiv 78 \pmod {125}$

Is there some relationship between modulo of powers of prime?

Answer 1

The answer is no. An easy counterexample try to find some $x$ such that $x \equiv 1 \mod 2$ and $x \equiv 2 \mod 4$.

The deeper reason for this is that we may state the CRT as

Let $q_1, ..., q_n$ be coprime, then \begin{equation*} \mathbb{Z}/ \left(\prod_{i=1}^n q_i \right) \cong \prod_{i=1}^n\mathbb{Z}/(q_i) \end{equation*}

However the same is not true when the $q_i$ are not coprime since $\mathbb{Z}/(p^n)$ is cyclic while $\prod_{i=1}^n\mathbb{Z}/(p)$ is not.

Fortunately it is not difficult to find $a \mod p^n$.