Prove that $p \ge 5$ is prime, then the remainder of $p$ upon division by $6$ is $1$ or $5$.

Question

An example in my textbook, but I'm not quite sure how to set this one up, because of the $p \ge 5$ part. How do I start it off?

Answer 1

If $p\ge 5$, then $p$ is odd. So, the remainder has to be odd. But, the remainder cannot be $3$ because $p\ge 5$ cannot be the form $6m+3$ which is divisible by $3$.