
\begin{enumerate}
\item $\begin{aligned}[t]
693 &= 1 \times 550 + 143\\[6pt]
 550& = 3 \times 143 + 121\\[6pt]
 143 &= 1 \times 121 + 22\\[6pt]
 121& = 5 \times  22 + 11\\[6pt]
  22 &= 2 \times  11 + 0.
\end{aligned}$

\medskip

Ainsi, $\pgcd{550}{693}=11$.

\item $\begin{aligned}[t]
11 & = 121 - 5\times22\\[6pt]
   & = (550 - 3\times143) - 5\times(143-121)\\[6pt]
   & = 550 - 3\times(693-550) - 5\big[(693-550) - 550 + 3(693-550)\big]\\[6pt]
   & = 550 - 3\times693 + 3\times550 - 5\times693 + 10\times550 - 15\times693 + 15\times550\\[6pt]
   & = 29\times550 - 23\times693.
\end{aligned}$

\medskip

Un couple \coord{u}{v} tel que $550u+693v=11$ est donc \coord{29}{-23}.
\end{enumerate}
