Let $p \in \mathbb{N}$ be prime with $p \gt 3$. Prove that $p^2 \equiv 1 \pmod {24}$.

Ideas I have so far:

- A prime must be odd, and an odd squared is odd
- Use induction?
- I know $24$ divides $1 - p^2$
- $1 + 24a = p^2$ where $p$ is a prime greater than $3$ and $a$ is an integer

I believe I have everything that I need to finish this problem, I just dont quite know how to put them together. Can anyone give me a tip/hint on how to finish? (An answer is not what I'm looking for :)) Thanks!