Let $L$ be a language consisting of a binary relation symbol $R$. Consider the following $L$-structures on $\mathbb N$:
$\mathcal N_1=(\mathbb N, \equiv_2),\quad \mathcal N_2=(\mathbb N,<),\quad\mathcal N_3=(\mathbb N,|),\quad\mathcal N_4=(\mathbb N,E)$
where $m\equiv_2 n$ iff $2|m-n$, $<$ is the usual ordering of natural numbers, $|$ is the usual divisibility relation, and $mEn$ iff $m$ and $n$ are consecutive and the smaller one is even. Show that any two of these structures are not elementarily equivalent to each other.
Definition: $\mathcal M \equiv \mathcal N$ if for every $L$-sentence $\sigma$, we have $\mathcal M\models \sigma \iff \mathcal N\models \sigma$
Take $\mathcal N_1 $ and $\mathcal N_2$:
Let $\varphi:=\exists x \exists y(x=3 \, \wedge \, y=4)$
Now, $\mathcal N_1 \nvDash \varphi$ since $2\nmid 3-4$ but $\mathcal N_2 \models \varphi$ since $3<4$.
So, they are not elementarily equivalent to each other.
I don't know how else argue this.