# Least Number with Three Given Fractions

## Theorem

In the words of Euclid:

(*The Elements*: Book $\text{VII}$: Proposition $39$)

## Proof

Let $a, b, c$ be the given aliquot parts.

Let $d, e, f$ be the numbers called by the same name as the aliquot parts $a, b, c$.

From Proposition $36$ of Book $\text{VII} $: LCM of Three Numbers, let:

- $g = \lcm \set {d, e, f}$

So $g$ has aliquot parts called by the same name as $d, e, f$.

Therefore $g$ has the aliquot parts $a, b, c$.

Suppose there exists $h \in \N: h < g$ which has the aliquot parts $a, b, c$.

By Proposition $38$ of Book $\text{VII} $: Divisor is Reciprocal of Divisor of Integer, $h$ will be measured by numbers called by the same name as the aliquot parts $a, b, c$.

Therefore $h$ is measured by $d, e, f$.

But $h < g$ which is impossible.

Therefore there is no number less than $g$ which has the aliquot parts $a, b, c$.

$\blacksquare$

## Historical Note

This proof is Proposition $39$ 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