(Endomorphisms of $\mathbb{Z}$) Find all endomorphisms of the group $(\mathbb{Z},+)$.
All applications $\varphi_{d}: x \rightarrow d x$, where $d$ is an integer, are indeed endomorphisms of $\mathbb{Z}$. Conversely, if $\varphi$ is an endomorphism of $\mathbb{Z}$, then we prove by induction on $|n|$ that, for any integer $n$, $\varphi(n)=n \varphi(1)$: this means that $\varphi=\varphi_{\varphi(1)}$. The endomorphisms of $\mathbb{Z}$ are therefore the linear functions with integer slope.