# Number multiplied by Cube Number making Cube is itself Cube

## 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.

## 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.

$a^3 \cdot a^3 : a^3 b = a^3 : b$

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.