# 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:

$a \equiv b \pmod {r s}$ if and only if $a \equiv b \pmod r$ and $a \equiv b \pmod s$

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

## Proof

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

We have that:

 $\ds 30$ $\equiv$ $\ds 40$ $\ds \pmod 2$ $\ds 30$ $\equiv$ $\ds 40$ $\ds \pmod {10}$ But note that: $\ds 30$ $\not \equiv$ $\ds 40$ $\ds \pmod {20}$

$\blacksquare$