Integer Divided by Divisor is Integer
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Theorem
Let $a, b \in \N$.
Then:
- $b \divides a \implies \dfrac 1 b \times a \in \N$
where $\divides$ denotes divisibilty.
In the words of Euclid:
- If a number be measured by any number, the number which is measured will have a part called by the same name as the measuring number.
(The Elements: Book $\text{VII}$: Proposition $37$)
Proof
Let $b \divides a$.
By definition of divisibilty:
- $\exists c \in \N: c \times b = a$
Then also:
- $c \times 1 = c$
So by Proposition $15$ of Book $\text{VII} $: Alternate Ratios of Multiples:
- $1 : b = c : a$
Hence the result.
$\blacksquare$
Historical Note
This proof is Proposition $37$ 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