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

$\blacksquare$