Square Modulo 24 of Odd Integer Not Divisible by 3/Proof 1

Proof
Let $a$ be as asserted.

We have that:
 * $2 \nmid a$

From Odd Square Modulo 8:


 * $a^2 \equiv 1 \pmod 8$

which means:
 * $8 \divides a^2 - 1$

We also have that:
 * $3 \nmid a$

From Square Modulo 3: Corollary 3:


 * $3 \divides a^2 - 1$

We have from Coprime Integers: $3$ and $8$ that:
 * $3 \perp 8$

where $\perp$ denotes coprimality.

As we have that:
 * $8 \divides a^2 - 1$

and:
 * $3 \divides a^2 - 1$

it follows from Product of Coprime Factors that:
 * $24 \divides a^2 - 1$

Hence the result.