If $xy = u^2 $and $yz = w^2$, where each variable is a natural number, why is$ (\frac{uw}{y}) $a natural number?

Question

I've just begun learning about equivalence relations, and this note was made in a worked example, but I'm unable to see why $(\frac{uw}{y}) $must also be a natural number. Could someone please provide a quick proof for me?

Thanks

Answer 1

$xy=u^2$

$yz=w^2$

$uw=y\sqrt{xz}$

As both sides are natural no.s $xz$ must be a perfect square, proving the result.

Answer 2

It follows from this Lemma: if $a$ and $b$ are natural numbers so that $b^2 | a^2$, then $b | a$.

In your example, $y^2 | (uw)^2$, therefore $y | uw$ by the Lemma.

There are many ways to prove the Lemma itself. For example, it obviously follows from the unique factorization theorem. Another way to refurmulate the lemma is like this: if the square root of a natural number is rational, then it is in fact natural.

Answer 3

Because its square $t^2=\frac{u^2w^2}{y^2}=xz$ is a natural number and $t$ is a rational number and every real number whose square is an integer is either an integer or an irrational number.