Number multiplied by Cube Number making Cube is itself 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.

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 Progressions 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 Progression is Cube then Fourth is Cube.