  We are familiar with properties and structure of topological spaces. One of
the powerful tools, which help us to figure out the structure of topological
spaces is (Leray- Serre) spectral sequence. Although Eilenberg-Maclane space
plays important roles in topology, and respectively geometry. Actually finding
cohomology groups of this space can be useful for classifying spaces, and also
homotopy groups structure of these groups. This paper discusses how to compute
cohomology groups of Eilenberg-Maclane spaces $K(\mathbb{Z}, 4)$ and
$K(\mathbb{Z}, 5)$ (cohomology degree less than $11$). Furthermore we give the
method to find cohomology groups of $K(\mathbb{Z}, n)$. Some proofs are given
to the basic facts about cohomology group of $K(\mathbb{Z}, 5)$ and
$K(\mathbb{Z},4).$
