prove that if $u$ is an integer which divides all integers, then $u$ is the unity

Question

prove that if $u$ is an integer which divides all integers, then $u$ is the unity

this is a divisibility exercise, i dont know how to use the definition of divisibility* and the definition of unity**

*$a\mid b \implies \exists c\in \mathbb Z$, such that, $b=a\cdot c$, whit $a,b\in \mathbb Z$

**if $u$ is a unity $\implies \exists u_{1}$, such that, $u\cdot u_{1}=1$

Answer 1

If $u$ divides every integer, then in particular $u$ divides $1$. So there is a $u_1$ such that $uu_1=1$.