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.

(The Elements: Book $\text{VIII}$: Proposition $15$)


Proof

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

From the corollary to Form of Geometric Progression of Integers:

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

is a geometric progression.

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

Then from First Element of Geometric Progression that divides Last also divides Second:

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

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.


Sources