Square Modulo 24 of Odd Integer Not Divisible by 3

Theorem
Let $a \in \Z$ be an integer such that:
 * $2 \nmid a$
 * $3 \nmid a$

where $\nmid$ denotes non-divisibility.

Then:
 * $a^2 \equiv 1 \pmod {24}$

That is:
 * $24 \divides \paren {a^2 - 1}$

where $\divides$ denotes divisibility.