Chinese Remainder Theorem/Warning

Theorem
Let $a, b, r, s \in \Z$. Let $r$ not be coprime to $s$.

Then it is not necessarily the case that:
 * $a \equiv b \pmod {r s}$ $a \equiv b \pmod r$ and $a \equiv b \pmod s$

where $a \equiv b \pmod r$ denotes that $a$ is congruent modulo $r$ to $b$.

Proof
Proof by Counterexample:

Let $a = 30, b = 40, r = 2, s = 10$.

We have that: