IMC 2002 Problem 9

For each $n\geq 1$ let $$a_{n}=\sum_{k=0}^{\infty}\frac{k^{n}}{k!}, \; b_{n}=\sum_{k=0}^{\infty}(-1)^{k}\frac{k^{n}}{k!}.$$ Show that $a_{n}\cdot b_{n}$ is an integer.