The Pattern in the Prime Squares

An analyst playing with primes notices that $5^2 - 1 = 24$, $7^2 - 1 = 48$, $11^2 - 1 = 120$, and $13^2 - 1 = 168$ are all multiples of $24$. Prove that this is no coincidence: for every prime $p > 3$, the number $p^2 - 1$ is divisible by $24$.