smc

Starting with some experimentation, substituting $p=5$ results in $24$, substituting $p=7$ results in $48$, and substituting $p=11$ results in $120$. For these primes, the resulting numbers are multiples of $24$. To show that all primes we devise the following proof: $Proof:$ $p\ge 5$ result in $p^2-1$ being a multiple of $24$, we can use modular arithmetic. Note that $p^2-1=(p+1)(p-1)$. Since $p^2\equiv 0,1,2,3\mathrm{mod}4$, $p^2-1$ is a multiple of $8$. Also, since $p^2\equiv 0,1,2\mathrm{mod}3$, $p^2-1$ is a multiple of $3$. Thus, $p^2-1$ is a multiple of $24$, so the answer is $\boxed{\text{always}}$.