Number divides Number iff Cube divides Cube

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Theorem

Let $a, b \in \Z$.

Then:

$a^3 \divides b^3 \iff a \divides b$

where $\divides$ 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 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:

\(\ds a^3\) \(\divides\) \(\ds a^2 b\)
\(\ds \leadsto \ \ \) \(\ds \exists k \in \Z: \, \) \(\ds k a^3\) \(=\) \(\ds a^2 b\) Definition of Divisor of Integer
\(\ds \leadsto \ \ \) \(\ds k a\) \(=\) \(\ds b\)
\(\ds \leadsto \ \ \) \(\ds a\) \(\divides\) \(\ds b\) Definition of Divisor of Integer

$\Box$


Let $a \divides b$.

Then:

\(\ds a\) \(\divides\) \(\ds b\)
\(\ds \leadsto \ \ \) \(\ds \exists k \in \Z: \, \) \(\ds k a\) \(=\) \(\ds b\) Definition of Divisor of Integer
\(\ds \leadsto \ \ \) \(\ds k a^3\) \(=\) \(\ds a^2 b\)
\(\ds \leadsto \ \ \) \(\ds a^3\) \(\divides\) \(\ds a^2 b\) Definition of Divisor of Integer
\(\ds \leadsto \ \ \) \(\ds a^3\) \(\divides\) \(\ds b^3\) Divisibility of Elements in Geometric Sequence of Integers

$\blacksquare$


Historical Note

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


Sources

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