<s>[INST] Show that if $p$ is a prime number $> 3$ then $24 \mid p^2-1$ Hi guys can someone help me with this ?(Without using Modular arithmetic)
Show that if $p$ is a prime number $>3$ then $24$ $\mid$ $p^2-1$
 [/INST] If $p$ is a prime greater than $3$, than it is equal to $1, 3, 5, 7$ $\pmod 8$
THis implies $p^2 \equiv 1 \pmod 8$ (because $1^2 \equiv 1$, $3^2 \equiv 1, 5^2 \equiv 1, 7^2 \equiv 1 \pmod 8$)
Also, $p \equiv 1, 2 \pmod 3 \Rightarrow p^2 \equiv 1 \pmod 3$
THis implies that $p^2 - 1 $ is divisible by both $8$ and $3$, and so it is divisible by $24$ 
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