Number divides Number iff Square divides Square

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Theorem

Let $a, b \in \Z$.

Then:

$a^2 \divides b^2 \iff a \divides b$

where $\divides$ denotes integer divisibility.


In the words of Euclid:

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

(The Elements: Book $\text{VIII}$: Proposition $14$)


Proof

From Between two Squares exists one Mean Proportional:

$\tuple {a^2, ab, b^2}$

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:

$a^2 \divides a b$

Thus:

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

$\blacksquare$


Historical Note

This proof is Proposition $14$ of Book $\text{VIII}$ of Euclid's The Elements.


Sources

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