# Odd Number minus Odd Number is Even

## Theorem

In the words of Euclid:

If from an odd number an odd number be subtracted, the remainder will be even.

## Proof

Let $a$ and $b$ be odd numbers.

Then by definition of odd number:

$\exists c \in \Z: a = 2 c + 1$
$\exists d \in \Z: b = 2 d + 1$

So:

 $\ds a - b$ $=$ $\ds 2 c + 1 - \left({2 d + 1}\right)$ $\ds$ $=$ $\ds 2 \left({c - d}\right)$

Hence the result by definition of even number.

$\blacksquare$

## Historical Note

This proof is Proposition $26$ of Book $\text{IX}$ of Euclid's The Elements.