Number divides Number iff Cube divides Cube

Theorem
Let $a, b \in \Z$.

Then:
 * $a^3 \divides b^3 \iff a \divides b$

where $\divides$ denotes integer divisibility.

Proof
Let $a^3$ and $b^3$ be cube numbers.

From the corollary to Form of Geometric Sequence of Integers:
 * $\tuple {a^3, a^2 b, a b^2, b^3}$

is a geometric sequence.

Let $a, b \in \Z$ such that $a^2 \divides b^2$.

Then from First Element of Geometric Sequence that divides Last also divides Second:
 * $a^3 \divides a^2 b$

Thus:

Let $a \divides b$.

Then: