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 $b$.

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:

$\ds 30$

$\not \equiv$

$\ds 40$

$\ds \pmod {20}$

$\blacksquare$