# The Even Numbers

An **even number** is an integer that can be exactly or evenly divided by \color{red}2. If a number is exactly divided by \color{red}2, it implies that the number in question has a remainder of \color{blue}0 upon the division of \color{red}2.

By performing mental math, it’s obvious that the numbers below, including the negative numbers, are even because they are all divisible by **2**.

In addition, I want to point out that many students think **zero** is neither even nor odd.

Believe me, zero is considered as an even number for the same simple reason that it is also an integer that is divisible by 2 thus no remainder when divided by 2 as well. That is, 0 \div 2 = 0.

**Observation:** From the examples above, we can easily generalize that even numbers always end with a digit of 0, 2, 4, 6, or 8.

However, there is a better way to define an even number because it’s more mathematically precise. Here it is!

## General Form of an Even Number

**DEFINITION:** The number \large{n} is an even number if we can express it as \large{2k} where \large{k} is just another integer. That means, \large{n} is even if \large{n = 2k} such that \large{k} is an integer.

## Examples of Even Numbers Expressed in General Form

Let’s put the general form of an even number to the test. It is important that whatever math concept we’re presented with, it is crucial that we verify it. It should somehow make sense to us before including it in our “math toolbox”.

Below are a few examples to showcase the concept of an even number as n = 2\,k where k is an integer.

\color{red}\LARGE {n = 2k}- 0 \to 0 = 2\left( 0 \right)

- 14 \to 14 = 2\left( 7 \right)

- - 32 \to - 32 = 2\left( { - 16} \right)

- 50 \to 50 = 2\left( {25} \right)

- - 78 \to - 78 = 2\left( { - 39} \right)

The constant appearance of 2 as one of the factors of an even number suggests that any even numbers are indeed a **multiple of 2**.

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