Let $\mathrm{P}$ be a real polynomial. The remainder of the Euclidean division of $P(X)$ by $X-1$ is 2 and the remainder of the Euclidean division by $X-2$ is 1. What is the remainder of the Euclidean division of $P(X)$ by $X^{2}-3 X+2 ?$
The key is to use the Chinese Remainder Theorem for polynomials: in this case, we introduce the polynomial $Q(X) = -X + 3$. Then $P(X)$ and $Q(X)$ have the same remainder in the Euclidean divisions by $X-1$ and $X-2$, which means that $P(X) - Q(X)$ is divisible by both polynomials $X-1$ and $X-2$, which are coprime, and thus by their product $X^2 - 3X + 2$. In particular, the sought remainder is $Q(X) = -X + 3$.