Cube Number multiplied by Cube Number is Cube
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Theorem
Let $a, b \in \N$ be natural numbers.
Let $a$ and $b$ be cube numbers.
Then $a b$ is also a cube number.
In the words of Euclid:
- If a cube number by multiplying a cube number make some number the product will be cube.
(The Elements: Book $\text{IX}$: Proposition $4$)
Proof
By the definition of cube number:
- $\exists r \in \N: r^3 = a$
- $\exists s \in \N: s^3 = b$
Thus:
\(\ds a b\) | \(=\) | \(\ds \paren {r^3} \paren {s^3}\) | Power of Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {r s}^3\) |
Thus:
- $\exists k = r s \in \N: a = k^3$
Hence the result by definition of cube number.
$\blacksquare$
Historical Note
This proof is Proposition $4$ of Book $\text{IX}$ of Euclid's The Elements.
The proof as given here is not that given by Euclid, as the latter is unwieldy and of limited use.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{IX}$. Propositions