Divisor is Reciprocal of Divisor of Integer

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