It is stated that in general, cannot reduce a congruence modulo a prime power into smaller moduli. What is the meaning implied by a prime power? What can be the reason for that?
This problem is stated in context of the statement below:
$a\equiv b \pmod 8 \implies a \equiv b \pmod4 \wedge a \equiv b \pmod2 $, but not converse.
I could not find a connect between the statement and the problem given.
On
Okay, I'll take a stab at this, but I'm still not sure what is being asked here.
I think the author is simply saying that the statement $a \equiv b $ mod $p^k$ is not equivalent to any statement of the form $a \equiv b$ mod $p^m$ where $m<k$.
The implication goes one direction: $a \equiv b$ mod $p^k$ implies that $a \equiv b$ mod $p^m$ for all $m<k$, but the converse is not true, as illustrated by the fact that $2 \equiv 12$ mod $5$ but $2 \not \equiv 12$ mod $25$.
$a\equiv b \pmod 8 \iff a \equiv b \pmod4 \wedge a \equiv b \pmod2$
implies that $\mathbb Z_8 \cong \mathbb Z_4 \times \mathbb Z_2$ and this is not true.
\begin{array}{|c|c|} \hline x & \mathbb Z_8 & \mathbb Z_4 & \mathbb Z_2 \\ \hline 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 0 \\ 3 & 3 & 3 & 1 \\ 4 & 4 & 0 & 0 \\ 5 & 5 & 1 & 1 \\ 6 & 6 & 2 & 0 \\ 7 & 7 & 3 & 1 \\ \hline \end{array}
Note, for example, that $3_8$ and $7_8$ in $\mathbb Z_8$ both correspond to $3_4 \in \mathbb Z_4$ and $1_2 \in \mathbb Z_2$