  In this paper, we determine an explicit formula for the number of walks in
$X_n = \textsf{Cay}(\mathbb{Z}_n,\mathbb{U}_n)$, the unitary Cayley Graphs of
order $n$, between any pair of its vertices. With this result, we give the
number of representations of a fixed residue class $\bmod{}n$ as the sum of $k$
units of $\mathbb{Z}_n$.
