Calculate for all integers $n \geq 3$ the GCD of $n^{3}-6 n^{2}+11 n-6$ and $n^{2}-4 n+4$.
We apply the desclide algorithm:

\[
\begin{aligned}
n^{3}-6 n^{2}+11 n-6 & =n\left(n^{2}-4 n+4\right)-2 n^{2}+7 n-6 \\
-2 n^{2}+7 n-6 & =-2\left(n^{2}-4 n+4\right)-n+2 \\
n^{2}-4 n+4 & =(n-2)(n-2)+0
\end{aligned}
\]

The GCD of \( n^{3}-6 n^{2}+11 n-6 \) and \( n^{2}-4 n+4 \) is then \( n-2 \).