Nonstandard model of the additive integers
= Nonstandard model of the additive integers
Compactness gives a countable model elementarily equivalent to $(\mathbb Z,+,0)$ that contains a nonzero element divisible by every positive standard integer. It is not isomorphic to $\mathbb Z$, so $\operatorname{Th}(\mathbb Z,+,0)$ is not aleph-zero-categorical.