The problem is following:

Given that $p$ is a prime number, $p > 3$. Prove that $(p^2 - 1)$ is divisible by $24$.

I started writing down the possible remainders of dividing $p$ by $24$ and got the following row: ${5; 7; 11; 13; 17; 19; 23}$

But am i even right at this point? If i am how do i prove that these are the only possible remainders?