I'm sure this is a simple question. But I want to know whether there is some error in my logic. I define prime numbers via:
$p\in\mathbb N\setminus\{1\}$ is prime if for every $a,b\in\mathbb N$ we have $p\mid ab\implies (p\mid a)\vee (p\mid b)$.
Now I want to show that each prime has exactly two factors. For this, I take some $n\in\mathbb N$ such that $n\mid p$. This means there exists some $k\in\mathbb N$ with $k\mid p$ such that $p=k\cdot n$. Hence we have $p\mid n\cdot k$ and since $p$ is prime this means $p\mid n$ or $p\mid k$.
If $p\mid n$ then $n\mid p$ means $n=p$. And if $p\mid k$ then again $k\mid p$ means $p=k$ and thus $n=1$.