= Solution
For the additive <first-order structure>, let
$$
\varphi_n(x):=\exists y\,(x=\underbrace{y+\cdots+y}_{n\text{ terms}}),\qquad n\geq1.
$$
It defines $n\mathbb Z$. These sets are pairwise distinct: if $m>n>0$, the integer $n$ belongs to $n\mathbb Z$ but not to $m\mathbb Z$. Thus there are infinitely many inequivalent one-variable <first-order formulas> modulo the <complete theory> of the additive integers. The <Ryll-Nardzewski theorem> implies \b[that <first-order theory> is not aleph-zero-categorical].
For the multiplicative <first-order structure> define divisibility by $d(y,x):=\exists z(yz=x)$. For $k\geq1$, the <first-order formula>
$$
\psi_k(x):=\exists y_1\cdots y_{2k}\left(\bigwedge_{i<j}y_i\ne y_j\ \land\ \bigwedge_{i=1}^{2k}d(y_i,x)\right)
$$
asserts the existence of at least $2k$ distinct integer divisors. The integer $2^{k-1}$ has exactly $2k$ such divisors, namely $\pm2^j$ for $0\leq j\leq k-1$, so it satisfies $\psi_k$ but not any $\psi_\ell$ with $\ell>k$. Again there are infinitely many inequivalent one-variable <first-order formulas>. The <Ryll-Nardzewski theorem> therefore implies \b[the <first-order theory> of the multiplicative integers is not aleph-zero-categorical].
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