[Math] Converse of the Chinese Remainder Theorem

Question

Is the converse of the Chinese Remainder Theorem true? That is, if
$$(m, n)\neq1,$$
then
$$\mathbb{Z}/mn\mathbb{Z}\ncong\mathbb{Z}/m\mathbb{Z}\oplus\mathbb{Z}/n\mathbb{Z}.$$

Thanks.

Answer 1

Yes. The direct sum has no element of order $mn$.