Section 3.7 Two facts from the gcd
Proposition 3.7.1. When perfect squares divide each other.
For integers \(a,z\) it is true that
\begin{equation*}
a^2\mid z^2\Longrightarrow a\mid z
\end{equation*}
Proof.
First, let \(d=\gcd(a,z)\text{.}\) Then we can write \(z^2=a^2 \cdot k\) for some integer \(k\text{,}\) and immediately write
\begin{equation*}
(z')^2 d^2=(a')^2 d^2 k
\end{equation*}
for some integers
\(z'\) and
\(a'\text{,}\) by definition of gcd. (That is,
\(z=z'd\) and
\(a=a'd\text{.}\) Also note that
\(z',a'\) are now relatively prime; it is not hard to prove using the techniques of the previous chapter, or see
Exercise 6.6.7.)
Cancelling the \(d^2\) (yes, we do assume this property of integers) yields
\begin{equation*}
(z')^2=(a')^2 k\text{.}
\end{equation*}
Since \(\gcd(a',z')=1\text{,}\) we have \(a'x+z'y=1\) for some \(x,y\in\mathbb{Z}\text{;}\) now we substitute for \(1\) in \(a'\cdot 1\cdot x\) (!) to get
\begin{equation*}
a'(a' x+z'y)x+z'y=1
\end{equation*}
Now we have that \(a'^2 x^2+z'(a'xy+y)=1\text{,}\) so that \(\gcd((a')^2,z')=1\) as well. But of course \(a'\mid (z')^2\text{.}\) Clearly if a positive number is a divisor, but their greatest common divisor is 1, then that number is going to have to be 1 by definition of divisors. So \(a'=1\text{.}\) (If \(a'\) was negative, the same argument for \(-a'\) shows \(-a'=1\text{,}\) so really \(a'=\pm 1\text{.}\))
Hence \(a=a'd=\pm d\text{,}\) which is a divisor of \(z\text{,}\) we have the desired result.
Proposition 3.7.2. When the product of coprime numbers is a square.
If we have integers \(m,n,j\) such that \(mn=j^2\) and \(\gcd(m,n)=1\text{,}\) then \(m\) and \(n\) are also both perfect squares.
Proof.
First, we will need a general fact about gcds:
\begin{equation*}
\gcd(x,y)^2=\gcd(x^2,xy,y^2)
\end{equation*}
We know that \(1=\gcd(m,n)=\gcd(m,n,j)\text{,}\) so
\begin{equation*}
m = m\cdot \gcd(m,n,j)=\gcd(m^2,mn,mj)=\gcd(m^2,j^2,mj)
\end{equation*}
Now we use the fact, so that
\begin{equation*}
m=\gcd(m,j)^2\text{.}
\end{equation*}
That’s a perfect square.
The same argument with \(n\) and \(j\) yields \(n = \gcd(n,j)^2\text{.}\)
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