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:

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

$\Box$


Let $a \divides b$.

Then:

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

$\blacksquare$


Historical Note

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


Sources