Odd Number multiplied by Even Number is Even

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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:

\(\displaystyle a \times b\) \(=\) \(\displaystyle \left({2 c + 1}\right) \times 2 d\)
\(\displaystyle \) \(=\) \(\displaystyle 2 \left({2 c d + d}\right)\)

Hence the result by definition of even number.

$\blacksquare$


Historical Note

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


Sources