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 theorem is Proposition $37$ of Book $\text{VII}$ of Euclid's The Elements.


Sources