KöMaL 2018 B4946

Let $f(x)$ be a polynomial of real coefficients such that $f(k)$ is an integer for every positive integer $k$ that ends in $5$ or $8$ in decimal notation. $a)$ Prove that $f(0)$ is an integer. $b)$ Give an example of a polynomial $f(x)$ that meets the above conditions, but $f(1)$ is not an integer.