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