Prime Vs Composite

How to prove if something is a prime or a composite (made up of a few smaller things?)

The definition you’re used to, complicated:

$n$ is prime $⟺$ $n>1$ and for all positive integers $r$ and $s$, if $n=rs$ then either $r$ or $s$ is $n$.

Basically:
n is prime iff one of the factors of n is n itself (and the other is 1 obviously).

Definition of Composite:

$n$ is composite $⟺$ $n>1$ and $n=rs$. For some integers $r$ and $s$ with $1<r<n$ and $1<s<n$.