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

Proof: By Euclid

(related to Proposition: 7.28: Numbers are Co-prime iff Sum is Co-prime to Both)

• "$\Rightarrow$"
• Let the two numbers, $AB$ and $BC$, (which are) prime to one another, be laid out.
  • Assume, $CA$ and $AB$ are not prime to one another.
  • Then some number will measure $CA$ and $AB$.
  • Let it (so) measure (them), and let it be $D$.
  • Therefore, since $D$ measures $CA$ and $AB$, it will thus also measure the remainder $BC$.
  • And it also measures $BA$.
  • Thus, $D$ measures $AB$ and $BC$, which are prime to one another.
  • The very thing is impossible.
  • Thus, some number cannot measure (both) the numbers $CA$ and $AB$.
  • Thus, $CA$ and $AB$ are prime to one another.
  • So, for the same (reasons), $AC$ and $CB$ are also prime to one another.
  • Thus, $CA$ is prime to each of $AB$ and $BC$.
• "$\Leftarrow$"
• So, again, let $CA$ and $AB$ be prime to one another.
  • For if $AB$ and $BC$ are not prime to one another then some number will measure $AB$ and $BC$.
  • Let it (so) measure (them), and let it be $D$.
  • And since $D$ measures each of $AB$ and $BC$, it will thus also measure the whole of $CA$.
  • And it also measures $AB$.
  • Thus, $D$ measures $CA$ and $AB$, which are prime to one another.
  • The very thing is impossible.
  • Thus, some number cannot measure (both) the numbers $AB$ and $BC$.
  • Thus, $AB$ and $BC$ are prime to one another.
• (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"