  Let $\Gamma=(V,E)$ be a graph of order $n$. A {\em closed distance magic
labeling} of $\Gamma$ is a bijection $\ell : V \to \{1,2, \ldots, n\}$ for
which there exists a positive integer $r$ such that $\sum_{x \in N[u]} \ell(x)
= r$ for all vertices $u \in V$, where $N[u]$ is the closed neighborhood of
$u$. A graph is said to be {\em closed distance magic} if it admits a closed
distance magic labeling.
  In this paper, we classify all connected closed distance magic circulants
with valency at most $5$, that is, Cayley graphs
$\operatorname{Cay}(\mathbb{Z}_n;S)$ where $|S| \le 5$ and $S$ generates
$\mathbb{Z}_n$.
