
$\pgcd{x}{y}=354 \iff \exists \coord{x'}{y'}\in\N,\ x=354x',\ y=354y',\ \pgcd{x'}{y'}=1$.

Ainsi, $x+y=\numprint{5664} \iff 354x'+354y'=\numprint{5664}\iff x'+y'=16$.

On cherche donc deux entiers $x'$ et $y'$ premiers entre eux dont la somme vaut 16. On peut en faire la liste:
\begin{itemize}
\begin{multicols}{4}
\item $\coord{x'}{y'}=\coord{1}{15}$
\item $\coord{x'}{y'}=\coord{3}{13}$
\item $\coord{x'}{y'}=\coord{5}{11}$
\item $\coord{x'}{y'}=\coord{7}{9}$
\item $\coord{x'}{y'}=\coord{9}{7}$
\item $\coord{x'}{y'}=\coord{11}{5}$
\item $\coord{x'}{y'}=\coord{13}{3}$
\item $\coord{x'}{y'}=\coord{15}{1}$
\end{multicols}
\end{itemize}
En multipliant ces valeurs par 354, on obtient tous les couples \coord{x}{y}:
\begin{itemize}
\begin{multicols}{2}
\item $\coord{x}{y}=\coord{354}{\numprint{5310}}$
\item $\coord{x}{y}=\coord{\numprint{1062}}{\numprint{4602}}$
\item $\coord{x}{y}=\coord{\numprint{1770}}{\numprint{3894}}$
\item $\coord{x}{y}=\coord{\numprint{2478}}{\numprint{3186}}$
\item $\coord{x}{y}=\coord{\numprint{3186}}{\numprint{2478}}$
\item $\coord{x}{y}=\coord{\numprint{3894}}{\numprint{1770}}$
\item $\coord{x}{y}=\coord{\numprint{4602}}{\numprint{1062}}$
\item $\coord{x}{y}=\coord{\numprint{5310}}{\numprint{354}}$
\end{multicols}
\end{itemize}
