24 divides Square of Odd Integer Not Divisible by 3 plus 23/Proof 1

Theorem

Let $a \in \Z$ be an integer such that:

$2 \nmid a$

$3 \nmid a$

where $\nmid$ denotes non-divisibility.

Then:

$24 \divides \paren {a^2 + 23}$

where $\divides$ denotes divisibility.

Proof

Let $a$ be as defined.

Then:

\(\ds 24\)

\(\divides\)

\(\ds \paren {a^2 - 1}\)

Square Modulo 24 of Odd Integer Not Divisible by 3

\(\ds \leadsto \ \ \)

\(\ds 24\)

\(\divides\)

\(\ds \paren {a^2 - 1 + 24}\)

\(\ds \leadsto \ \ \)

\(\ds 24\)

\(\divides\)

\(\ds \paren {a^2 + 23}\)

$\blacksquare$