# Odd Number multiplied by Odd Number is Odd

## Theorem

In the words of Euclid:

If an odd number by multiplying an odd number make some number, the product will be odd.

## Proof

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

Then by definition of odd number:

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

So:

 $\ds a \times b$ $=$ $\ds \paren {2 c + 1} \times \paren {2 d + 1}$ $\ds$ $=$ $\ds 4 c d + 2 c + 2 d + 1$ $\ds$ $=$ $\ds 2 \paren {2 c d + c + d} + 1$

Hence the result by definition of odd number.

$\blacksquare$

## Historical Note

This proof is Proposition $29$ of Book $\text{IX}$ of Euclid's The Elements.