Anyone wanna prove this?

Every prime number except 2 and 3 has the form 6q + 1 or 6q + 5 for some integer q. Hint: Use the quotient-remainder theorem to say that n must have one of the forms 6q, 6q + 1, 6q + 2, 6q + 3, 6q + 4, or 6q + 5 for some integer q.

I can never make progress on these damn problems, it seems the only way to do them is if you already know the answer.

Seriously, does that prove that all integers have the form 6q + 1 or 6q + 5? I can think of ones that don't, for instance 6 is of the form 6q.

I try to do my own homework, really, but neither the book nor the teacher is of any help on half of the problems assigned, usually. I have a test this week and I need an understanding of the problems I had to skip.