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

Proof: By Euclid

(related to Proposition: Prop. 9.29: Odd Number multiplied by Odd Number is Odd)

• For let the odd number $A$ make $C$ (by) multiplying the odd (number) $B$.
• I say that $C$ is odd.

• For since $A$ has made $C$ (by) multiplying $B$, $C$ is thus composed out of so many (magnitudes) equal to $B$, as many as (there) are units in $A$ [Def. 7.15] .
• And each of $A$, $B$ is odd.
• Thus, $C$ is composed out of odd (numbers), (and) the multitude of them is odd.
• Hence $C$ is odd [Prop. 9.23].
• (Which is) the very thing it was required to show.
  ∎

Thank you to the contributors under CC BY-SA 4.0!

Github:

non-Github:

@Fitzpatrick

References

Adapted from (subject to copyright, with kind permission)

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