Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Discrete mathematics, continuous mathematics, and probability / 2 / b / i / Solution

Suppose, for a proof by contradiction, that an integer $n$ is both an even number and an odd number. Then $n=2u$ and $n=2v+1$ for some integers $u,v$. Subtracting yields $u-v=1/2$. The left side is an integer, while $1/2$ is not. This contradiction proves that no integer is both even and odd.