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

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.

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.

$\blacksquare$