Let $ n \in \mathbb{N} $. Prove that if $n \not\equiv 1$ (mod $6$) and $n \not\equiv 5 $(mod $6$), then $n$ is not prime or $n = 2$ or $ n = 3$.
We can prove this using the contrapositive.
It is true when $ n = 1 $ and $n = 5$. As $ 1 \equiv 1 (mod 6)$ and $ 5 \equiv 5 (mod 6) $. Similarly, it is also true for 7 and 11. How can I prove it is true for all primes?