# Even and odd numbers are used in arithmetic

Even and odd numbers are used in arithmetic, especially in the proof of properties. In this post, we show how to prove that a natural number is odd or even. Before doing this, let us give some definitions.

We say that a natural number $a$ is even if it can be written as $a=2k$ where $k$ is a natural number. On the other hand, $a$ is called an odd number if we can write $a$ as $a=2k+1,$ where $k$ is a natural number. The following are important properties of this kind of numbers:

• The sum, the difference and the product of two even numbers is still an even number.
• The sum or the difference of an even number with another odd number is an odd number
• The product of an even number with another odd number is an even number

### How to prove that a natural number is even or odd

Let $n$ be a natural number. We introduce three examples in which we prove the parity of numbers.

Let $a=6n+3$. Remark that $a=6n+2+1=2(3n+1)+1=2k+1$, with $k=3n+1\in \mathbb{N}$. Thus $a$ is odd.

Let $b=4n+6$. Immediately we observe that $b=2(2n+3)=2k,$ where $k=2n+3\in\mathbb{N}$. Thus $b$ is even.

Let $c=(2n+1)^2+2n-1$. We can write

\begin{align*}c&=(2n+1)^2+2n-1\cr &= 4n^2+4n+1+2n-1\cr &= 4n^2+6n\cr &= 2(2n^2+3n).\end{align*} This shows that $c$ is an even number.