Divisor is Reciprocal of Divisor of Integer
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Theorem
Let $a, b, c \in \Z_{>0}$.
Then:
- $b = \dfrac 1 c \times a \implies c \divides a$
where $\divides$ denotes divisibilty.
In the words of Euclid:
- If a number have any part whatever, it will be measured by a number called by the same name as the part.
(The Elements: Book $\text{VII}$: Proposition $38$)
Proof
Let $a$ have an aliquot part $b$.
Let $c$ be an integer called by the same name as the aliquot part $b$.
Then:
- $1 = \dfrac 1 c \times c$
and so by Proposition $15$ of Book $\text{VII} $: Alternate Ratios of Multiples:
- $ 1 : c = b : a$
Hence the result.
$\blacksquare$
Historical Note
This proof is Proposition $38$ of Book $\text{VII}$ 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 $\text{VII}$. Propositions