Elementary Methods in Number Theory Exercise 1.2.16

Consider $(\frac{25}{6},-5,\frac{10}{3})\in\mathbf{Q}^3$.Find all triples $(a_0,a_{1},a_2)$ of relatively prime integers such that
\begin{equation}
(a_0,a_1,a_2)\sim (\frac{25}{6},-5,\frac{10}{3})
\end{equation}

Solve:
\begin{align*}
\begin{cases}
a_0=t \frac{25}{6}\\
a_1=-5t\\
a_2=t \frac{10}{3}\\
\end{cases}
\end{align*}
$t\neq 0$.Let $t=\frac{6}{5}k$.
\begin{align*}
\begin{cases}
a_0=5k\\
a_1=-6k\\
a_2=4k\\
\end{cases}
\end{align*}
Let $k=\pm 1$,done.