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 $mn$.

Answer 2

HINT $\rm\ \ \mathbb Z/m\: \oplus\: \mathbb Z/n\ $ has characteristic $\rm\:lcm(m,n),\:$ which is $\rm\: < m\ n\ $ if $\rm\:\gcd(m,n) > 1\:.$