If Ratio of Cube to Number is as between Two Cubes then Number is Cube

Theorem
Let $a, b, c, d \in \Z$ be integers such that:
 * $\dfrac a b = \dfrac {c^3} {d^3}$

Let $a$ be a cube number.

Then $b$ is also a cube number.

Proof
From :
 * $\left({c^3, c^2 d, c d^2, d^3}\right)$

is a geometric progression.

From :
 * $\left({a, m_1, m_2, b}\right)$

is a geometric progression for some $m$.

We have that $a$ is a cube number.

From :
 * $b$ is a cube number.