Odd Divisor of Even Number also divides its Half
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Theorem
Let $a, b \in \Z$ be integers.
Let $a$ be odd and $b$ be even.
Let:
- $a \divides b$
where $\divides$ denotes divisibility.
Then:
- $a \divides \dfrac b 2$
In the words of Euclid:
- If an odd number measure an even number, it will also measure the half of it.
(The Elements: Book $\text{IX}$: Proposition $30$)
Proof
By definition of an even number:
- $\exists r \in \Z: b = 2 r$
By definition of an odd number:
- $\exists s \in \Z: a = 2 s + 1$
Thus:
- $a \divides 2 \iff a = \pm 1$
in which case from One Divides all Integers:
- $a \divides \dfrac b 2$
We have that:
- $a \divides 2 r$
and so by Euclid's Lemma:
- $a \divides r = \dfrac b 2$
$\blacksquare$
Historical Note
This proof is Proposition $30$ 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