Def/Even

Definition of Even: An integer $x$ is called even if it is a multiple of $2$. In other words, the following two statements are equivalent:

• $x$ is an even integer.
• There exists an integer $y$, such that $x = 2y$.

For example, $12$ is an even integer, since $12 = 2 \cdot 6$ and $6$ is an integer. Also, $-10$ is an even integer, since $-10 = 2 \cdot (-5)$ (and $-5$ is an integer). Also, $0$ is an even integer, since $0 = 2 \cdot 0$ (and $0$ is an integer).

On the other hand, $3$ is not an even integer; there does not exist an integer $y$ such that $3 = 2 \cdot y$.

A few useful properties of even integers are the following:

• The sum or difference of two even integers is again an even integer.
• The product of any integer with an even integer is again an even integer.
  • In particular, the product of any two even integers is an even integer.