Number divides Number iff Cube divides Cube

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Theorem

Let $a, b \in \Z$.

Then:

$a^3 \mathrel \backslash b^3 \iff a \mathrel \backslash b$

where $\backslash$ denotes integer divisibility.

In the words of Euclid:

If a cube number measure a cube number, the side will also measure the side; and, if the side measure the side, the cube will also measure the cube.

Proof

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

$\left({a^3, a^2 b, a b^2, b^3}\right)$

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

$a^3 \mathrel \backslash a^2 b$

Thus:

 $\displaystyle a^3$ $\backslash$ $\displaystyle a^2 b$ $\displaystyle \implies \ \$ $\, \displaystyle \exists k \in \Z: \,$ $\displaystyle k a^3$ $=$ $\displaystyle a^2 b$ Definition of Divisor of Integer $\displaystyle \implies \ \$ $\displaystyle k a$ $=$ $\displaystyle b$ $\displaystyle \implies \ \$ $\displaystyle a$ $\backslash$ $\displaystyle b$ Definition of Divisor of Integer

$\Box$

Let $a \mathrel \backslash b$.

Then:

 $\displaystyle a$ $\backslash$ $\displaystyle b$ $\displaystyle \implies \ \$ $\, \displaystyle \exists k \in \Z: \,$ $\displaystyle k a$ $=$ $\displaystyle b$ Definition of Divisor of Integer $\displaystyle \implies \ \$ $\displaystyle k a^3$ $=$ $\displaystyle a^2 b$ $\displaystyle \implies \ \$ $\displaystyle a^3$ $\backslash$ $\displaystyle a^2 b$ Definition of Divisor of Integer $\displaystyle \implies \ \$ $\displaystyle a^3$ $\backslash$ $\displaystyle b^3$ Divisibility of Elements in Geometric Progression of Integers

$\blacksquare$

Historical Note

This theorem is Proposition $15$ of Book $\text{VIII}$ of Euclid's The Elements.