Why are primes of the form $6k+1, 6k-1$ where the prime is $\geq 3$

Question

I recently came to know that primes are of form $6k+1,6k-1$ for primes greater than three. Why is this so? I tried my hand on it could not really understand about it. I have also heard of Dirichlet's theorem but can there be any elementary such way to show this?

Answer 1

What are the other possibilities? Numbers of the form $6k$, $6k+2$, or $6k+4$ are all divisible by $2$, while numbers of the form $6k+3$ are divisible by $3$ and thus are not prime if $k>0$.

Answer 2

$6k,6k+2,6k+3,6k+4$ cannot be prime for $k>0$ because they are divisible by $6,2,3,2$ respectively. Thus only $6k+1$ and $6k+5$ are left.