◀ ▲ ▶Branches / Geometry / Elements-euclid / Book--7-elementary-number-theory / Proof: By Euclid

Proof: By Euclid

(related to Proposition: 7.03: Greatest Common Divisor of Three Numbers)

• For let the greatest common measure, $D$, of the two (numbers) $A$ and $B$ have been taken [Prop. 7.2].
• So $D$ either measures, or does not measure, $C$.
• First of all, let it measure ($C$).
• And it also measures $A$ and $B$.
• Thus, $D$ measures $A$, $B$, and $C$.
• Thus, $D$ is a common measure of $A$, $B$, and $C$.
• So I say that (it is) also the greatest (common measure).
• For if $D$ is not the greatest common measure of $A$, $B$, and $C$ then some number greater than $D$ will measure the numbers $A$, $B$, and $C$.
• Let it (so) measure ($A$, $B$, and $C$), and let it be $E$.
• Therefore, since $E$ measures $A$, $B$, and $C$, it will thus also measure $A$ and $B$.
• Thus, it will also measure the greatest common measure of $A$ and $B$ [Prop. 7.2 corr.] .
• And $D$ is the greatest common measure of $A$ and $B$.
• Thus, $E$ measures $D$, the greater (measuring) the lesser.
• The very thing is impossible.
• Thus, some number which is greater than $D$ cannot measure the numbers $A$, $B$, and $C$.
• Thus, $D$ is the greatest common measure of $A$, $B$, and $C$.
• So let $D$ not measure $C$.
• I say, first of all, that $C$ and $D$ are not prime to one another.
• For since $A$, $B$, $C$ are not prime to one another, some number will measure them.
• So the (number) measuring $A$, $B$, and $C$ will also measure $A$ and $B$, and it will also measure the greatest common measure, $D$, of $A$ and $B$ [Prop. 7.2 corr.] .
• And it also measures $C$.
• Thus, some number will measure the numbers $D$ and $C$.
• Thus, $D$ and $C$ are not prime to one another.
• Therefore, let their greatest common measure, $E$, have been taken [Prop. 7.2].
• And since $E$ measures $D$, and $D$ measures $A$ and $B$, $E$ thus also measures $A$ and $B$.
• And it also measures $C$.
• Thus, $E$ measures $A$, $B$, and $C$.
• Thus, $E$ is a common measure of $A$, $B$, and $C$.
• So I say that (it is) also the greatest (common measure).
• For if $E$ is not the greatest common measure of $A$, $B$, and $C$ then some number greater than $E$ will measure the numbers $A$, $B$, and $C$.
• Let it (so) measure ($A$, $B$, and $C$), and let it be $F$.
• And since $F$ measures $A$, $B$, and $C$, it also measures $A$ and $B$.
• Thus, it will also measure the greatest common measure of $A$ and $B$ [Prop. 7.2 corr.] .
• And $D$ is the greatest common measure of $A$ and $B$.
• Thus, $F$ measures $D$.
• And it also measures $C$.
• Thus, $F$ measures $D$ and $C$.
• Thus, it will also measure the greatest common measure of $D$ and $C$ [Prop. 7.2 corr.] .
• And $E$ is the greatest common measure of $D$ and $C$.
• Thus, $F$ measures $E$, the greater (measuring) the lesser.
• The very thing is impossible.
• Thus, some number which is greater than $E$ does not measure the numbers $A$, $B$, and $C$.
• Thus, $E$ is the greatest common measure of $A$, $B$, 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)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"