even number

Definition Suppose $k$ is an integer. If there exists an integer $r$ such that $k=2r+1$ , then $k$ is an odd number. If there exists an integer $r$ such that $k=2r$ , then $k$ is an even number.

The concept of even and odd numbers are most easily understood in the binary base. Then the above definition simply states that even numbers end with a $0$ , and odd numbers end with a $1$ .

Properties

1. Every integer is either even or odd. This can be proven using induction, or using the fundamental theorem of arithmetic.
2. An integer $k$ is even (odd) if and only if $k^2$ is even (odd).