We introduce a basis of rational polynomial-like functions $P_0,\ldots,P_{n-1}$ for the free module of functions $\Z/n\Z\to\Z/m\Z$. We then characterize the subfamily of congruence preserving functions as the set of linear combinations of the functions $\lcm(k)\,P_k$ where $\lcm(k)$ is the least common multiple of $2,\ldots,k$ (viewed in $\Z/m\Z$). As a consequence, when $n\geq m$, the number of such functions is independent of $n$.