Odd Number multiplied by Even Number is Even

Theorem

In the words of Euclid:

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

(The Elements: Book $\text{IX}$: Proposition $28$)

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:

\(\ds a \times b\)

\(=\)

\(\ds \left({2 c + 1}\right) \times 2 d\)

\(\ds \)

\(=\)

\(\ds 2 \left({2 c d + d}\right)\)

Hence the result by definition of even number.

$\blacksquare$

Historical Note

This proof is Proposition $28$ of Book $\text{IX}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{IX}$. Propositions