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.

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