Let $P$ be a real polynomial such that $P(2)=5$ and $P(1)=3$. Determine the remainder of the Euclidean division of $P$ by $(X-2)(X-1)$.
Let $R$ be the remainder of the Euclidean division of $P$ by $(X-2)(X-1)$. There exists a polynomial $Q$ such that $P(X)=Q(X)(X-1)(X-2)+R(X)$. Moreover, the degree of $R$ is strictly less than the degree of $(X-2)(X-1)$, which is 2. Therefore, we can write $R$ in the form $aX+b$.

By evaluating $P(X)=Q(X)(X-1)(X-2)+R(X)$ at 1 and 2, we get $5=R(2)=2a+b$ and $3=R(1)=a+b$. By subtracting, we obtain $a=2$, and from $a+b=3$, we deduce $b=1$. Thus, the remainder is $2X+1$.