Putnam 2004 B1

Let $P(x) = c_n x^n + c_{n-1} x^{n-1} + \cdots + c_0$ be a polynomial with integer coefficients. Suppose that $r$ is a rational number such that $P(r) = 0$. Show that the $n$ numbers

$$\begin{gathered}c_n r, c_n r^2 + c_{n-1} r, c_n r^3 + c_{n-1} r^2 + c_{n-2} r, \\ \dots, c_n r^n + c_{n-1} r^{n-1} + \cdots + c_1 r\end{gathered}$$

are integers.