[proofplan] We divide the given Pythagorean triple by the greatest common divisor of its three entries. Since this divisor divides each coordinate, the quotient triple is integral. Substituting the quotient into the Pythagorean equation shows that the quotient triple is again Pythagorean. Finally, any common divisor of the quotient coordinates would enlarge the original greatest common divisor, so the quotient triple is primitive. [/proofplan] [step:Divide the triple by its greatest common divisor] Let \begin{align*} d := \gcd(|x|,|y|,|z|). \end{align*} Since $(x,y,z) \neq (0,0,0)$, at least one of $|x|$, $|y|$, and $|z|$ is positive, so $d \in \mathbb{N}$. By the definition of greatest common divisor, $d$ divides $x$, $d$ divides $y$, and $d$ divides $z$. Therefore there exist integers $x_0,y_0,z_0 \in \mathbb{Z}$ such that \begin{align*} x = d x_0, \qquad y = d y_0, \qquad z = d z_0. \end{align*} Equivalently, \begin{align*} (x,y,z) = d(x_0,y_0,z_0). \end{align*} [/step] [step:Check that the quotient triple is still Pythagorean] Substitute $x = d x_0$, $y = d y_0$, and $z = d z_0$ into the equation $x^2 + y^2 = z^2$. This gives \begin{align*} (d x_0)^2 + (d y_0)^2 &= (d z_0)^2, \\ d^2 x_0^2 + d^2 y_0^2 &= d^2 z_0^2. \end{align*} Since $d \in \mathbb{N}$, we have $d^2 \neq 0$, and division by $d^2$ gives \begin{align*} x_0^2 + y_0^2 = z_0^2. \end{align*} Thus $(x_0,y_0,z_0)$ is a Pythagorean triple. [/step] [step:Show that the quotient triple has no nontrivial common divisor] Let \begin{align*} e := \gcd(|x_0|,|y_0|,|z_0|). \end{align*} Then $e$ divides $x_0$, $y_0$, and $z_0$. Hence $de$ divides $d x_0 = x$, $d y_0 = y$, and $d z_0 = z$. Therefore $de$ is a common divisor of $|x|$, $|y|$, and $|z|$. Because $d$ is the greatest common divisor of $|x|$, $|y|$, and $|z|$, every common divisor of these three integers is at most $d$. Thus \begin{align*} de \le d. \end{align*} Since $d \in \mathbb{N}$, division by $d$ gives $e \le 1$. Also $e \in \mathbb{N}$, so $e = 1$. Therefore \begin{align*} \gcd(|x_0|,|y_0|,|z_0|) = 1. \end{align*} The quotient triple is primitive, and the original triple is the integer multiple $d(x_0,y_0,z_0)$ of that primitive Pythagorean triple. [/step]