◀ ▲ ▶Branches / Geometry / Elements-euclid / Book--9-number-theory-applications / Proof: By Euclid

Proof: By Euclid

(related to Proposition: Prop. 9.31: Odd Number Co-prime to Number is also Co-prime to its Double)

• For let the odd number $A$ be prime to some number $B$.
• And let $C$ be double $B$.
• I say that $A$ is [also] [prime to]bookofproofs$1288 $C$.

• For if [$A$ and $C$] are not prime (to one another) then some number will measure them.
• Let it measure (them), and let it be $D$.
• And $A$ is odd.
• Thus, $D$ (is) also odd.
• And since $D$, which is odd, measures $C$, and $C$ is even, [$D$] will thus also measure half of $C$ [Prop. 9.30].
• And $B$ is half of $C$.
• Thus, $D$ measures $B$.
• And it also measures $A$.
• Thus, $D$ measures (both) $A$ and $B$, (despite) them being prime to one another.
• The very thing is impossible.
• Thus, $A$ is not unprime to $C$.
• Thus, $A$ and $C$ 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"