How to split congruences so moduli are prime powers?

Question

If I have the linear congruence x=5 mod 84, is this equal to x=2 mod 3, since 3|84? This seems too easy.

Answer 1

$84 = 3 \times 2^2 \times 7$

By the Chines remainer theorem

$x \equiv 5 \pmod{84} \iff x\equiv 2 \pmod 3 \wedge x\equiv 1 \pmod 4 \wedge x \equiv 5 \pmod 7$

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Doing it the long way...

\begin{align} x \equiv 5 \pmod {84} &\iff x - 5 = 84n \; \text{for some $n \in \mathbb Z$}\\ &\iff x - 2 = 84n - 3 \; \text{for some $n \in \mathbb Z$}\\ &\iff x - 2 = 3(28n - 1) \; \text{for some $n \in \mathbb Z$}\\ &\implies x \equiv 2 \pmod 3\\ \end{align}