Odd Number multiplied by Even Number is Even
Jump to navigation
Jump to search
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