# 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