# Even Number minus Odd Number is Odd

## Theorem

In the words of Euclid:

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

## Proof

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

Then by definition of even number:

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

and by definition of odd number:

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

So:

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

Hence the result by definition of odd number.

$\blacksquare$

## Historical Note

This proof is Proposition $25$ of Book $\text{IX}$ of Euclid's The Elements.