Is that true that all the prime numbers are of the form $6m \pm 1$?
Q. Why is it that all primes greater than 3 are either 1 or -1 modulo 6?
Does it suffice to argue as follows:
Let $p$ be a prime. $p>3 \Rightarrow 3$ does not divide $p$. Clearly $2$ does not divide $p$ either, and so 6 does not divide p.
Now, $p$ is odd, and so $p$ is either 1,3 or 5 modulo 6. However, if $p$ were 3 (mod6), that would give us that $3$ divides $p$, which is a contradiction.
As such, we conclude that $p$ is either 1 or 5 (=-1) mod 6