  A simple graph on $n$ vertices may contain a lot of maximum cliques. But how
many can it potentially contain? We will define prime and composite graphs, and
we will show that if $n \ge 15$, then the grpahs with the maximum number of
maximum cliques have to be composite. Moreover, we will show an edge bound from
which we will prove that if any factor of a composite graph has $\omega(G_i)
\ge 5$, then it cannot have the maximum number of maximum cliques. Using this
we will show that the graph that contains $3^{\lfloor n/3 \rfloor}c$ maximum
cliques has the most number of maximum cliques on $n$ vertices, where
$c\in\{1,\frac{4}{3},2\}$, depending on $n \text{ mod } 3$.
