# Odd Number minus Even Number is Odd

## Theorem

In the words of Euclid:

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

## Proof

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

Then by definition of odd number:

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

and by definition of even number:

$\exists d \in \Z: b = 2 d$

So:

 $\ds a - b$ $=$ $\ds 2 c + 1 - 2 d$ $\ds$ $=$ $\ds 2 \paren {c - d} + 1$

Hence the result by definition of odd number.

$\blacksquare$

## Historical Note

This proof is Proposition $27$ of Book $\text{IX}$ of Euclid's The Elements.