Number multiplied by Cube Number making Cube is itself Cube
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Theorem
In the words of Euclid:
- If a cube number by multiplying any number make a cube number, the multiplied number will also be cube.
(The Elements: Book $\text{IX}$: Proposition $5$)
Proof
Let $a, b \in \Z$ be integers such that $a^3 b$ is a cube number.
By Square of Cube Number is Cube, $a^3 \cdot a^3$ is a cube number.
By Multiples of Ratios of Numbers:
- $a^3 \cdot a^3 : a^3 b = a^3 : b$
From Between two Cubes exist two Mean Proportionals, there exist two mean proportionals between $a^3 \cdot a^3$ and $a^3 b$.
By Geometric Sequences in Proportion have Same Number of Elements, there exist two mean proportionals between $a^3$ and $b$.
The result follows from If First of Four Numbers in Geometric Sequence is Cube then Fourth is Cube.
$\blacksquare$
Historical Note
This proof is Proposition $5$ of Book $\text{IX}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{IX}$. Propositions