Solution
(related to Problem: Checking if $K_5$ is Planar)
 Suppose that the complete graph $K_5$ is planar.
 A necessary condition for a graph to be planar is that $E\le 3V6,$ where $E$ ist the number of its edges and $V$ is the number of its vertices.
 For $K_5$ this condition means that $$E=10\le 3V6=3\cdot 56=9.$$ which is false.
 Therefore, $K_5$ is not planar.
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References
Bibliography
 Aldous Joan M., Wilson Robin J.: "Graphs and Applications  An Introductory Approach", Springer, 2000