Proof: By Euclid
(related to Proposition: 7.36: Least Common Multiple of Three Numbers)
- For let the least (number), $D$, measured by the two (numbers) $A$ and $B$ have been taken [Prop. 7.34].
- So $C$ either measures, or does not measure, $D$.
- Let it, first of all, measure ($D$).
- And $A$ and $B$ also measure $D$.
- Thus, $A$, $B$, and $C$ (all) measure $D$.
- So I say that ($D$ is) also the least (number measured by $A$, $B$, and $C$).
- For if not, $A$, $B$, and $C$ will (all) measure [some] number which is less than $D$.
- Let them measure $E$ (which is less than $D$).
- Since $A$, $B$, and $C$ (all) measure $E$ then $A$ and $B$ thus also measure $E$.
- Thus, the least (number) measured by $A$ and $B$ will also measure [$E$] [[Prop. 7.35]]bookofproofs$2365.
- And $D$ is the least (number) measured by $A$ and $B$.
- Thus, $D$ will measure $E$, the greater (measuring) the lesser.
- The very thing is impossible.
- Thus, $A$, $B$, and $C$ cannot (all) measure some number which is less than $D$.
- Thus, $A$, $B$, and $C$ (all) measure the least (number) $D$.
- So, again, let $C$ not measure $D$.
- And let the least number, $E$, measured by $C$ and $D$ have been taken [Prop. 7.34].
- Since $A$ and $B$ measure $D$, and $D$ measures $E$, $A$ and $B$ thus also measure $E$.
- And $C$ also measures [$E$].
- Thus, $A$, $B$, and $C$ [also] measure $E$.
- So I say that ($E$ is) also the least (number measured by $A$, $B$, and $C$).
- For if not, $A$, $B$, and $C$ will (all) measure some (number) which is less than $E$.
- Let them measure $F$ (which is less than $E$).
- Since $A$, $B$, and $C$ (all) measure $F$, $A$ and $B$ thus also measure $F$.
- Thus, the least (number) measured by $A$ and $B$ will also measure $F$ [Prop. 7.35].
- And $D$ is the least (number) measured by $A$ and $B$.
- Thus, $D$ measures $F$.
- And $C$ also measures $F$.
- Thus, $D$ and $C$ (both) measure $F$.
- Hence, the least (number) measured by $D$ and $C$ will also measure $F$ [Prop. 7.35].
- And $E$ is the least (number) measured by $C$ and $D$.
- Thus, $E$ measures $F$, the greater (measuring) the lesser.
- The very thing is impossible.
- Thus, $A$, $B$, and $C$ cannot measure some number which is less than $E$.
- Thus, $E$ (is) the least (number) which is measured by $A$, $B$, and $C$.
- (Which is) the very thing it was required to show.
∎
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References
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
