  The unitary Cayley graph of $\mathbb Z/n\mathbb Z$, denoted $G_{\mathbb
Z/n\mathbb Z}$, is the graph with vertices $0,1,\ldots,$ $n-1$ in which two
vertices are adjacent if and only if their difference is relatively prime to
$n$. These graphs are central to the study of graph representations modulo
integers, which were originally introduced by Erd\H{o}s and Evans. We give a
brief account of some results concerning these beautiful graphs and provide a
short proof of a simple formula for the number of cliques of any order $m$ in
the unitary Cayley graph $G_{\mathbb Z/n\mathbb Z}$. This formula involves an
exciting class of arithmetic functions known as Schemmel totient functions,
which we also briefly discuss. More generally, the proof yields a formula for
the number of cliques of order $m$ in a direct product of balanced complete
multipartite graphs.
