# Even And Odd Numbers: Rules And Properties

The set of integers consists of two types of numbers, the even numbers and the odd numbers. In this lesson, we will be able to identify the various properties that distinguish these numbers due to their great importance in the field of mathematics, especially arithmetic and number theory.

Even and odd numbers form the basic building block for understanding how numbers are distributed within the set of integers, as well as a basic starting point for studying prime numbers.

## Even Numbers

An even number is any number that is divisible by 2, that is, the remainder of dividing this number by 2 is 0.

Here are some examples of even numbers:

For the number , it is also an even number because , so the even number rule applies to it.

Now note carefully these even numbers:

You will no doubt have noticed that even numbers always end in one of the digits or on the far right

### The General Formula For Even Numbers

Let's take the following even numbers:

You can see that an even number can always be decomposed into a product, one of whose factors is 2.

### Note:

We say that is an even number if it can be written as:

where is an integer.

## Odd Numbers

An odd Number is any number that is not divisible by 2, that is, the remainder of dividing this number by 2 is 1.

Here are some examples of odd numbers:

Now note carefully these odd numbers:

You will no doubt have noticed that odd numbers always end in one of the digits or on the far right

## The General Formula For Odd Numbers

Let's take the following odd numbers:

You can see that an odd number can always be decomposed into a sum of 1 and an even number.

### Note:

We say that is an odd number if it can be written as:

where is an integer.

#### Example1:

Write the following odd numbers as odd number form:

Solution

The form of odd number is , so we must fin the integer .

## Properties of Even and Odd Numbers

In this section, we will try to identify some important properties of even and odd numbers through the use of some algebraic properties. Each property we will demonstrate with a simple proof.

### Note:

The sum of two even numbers is an even number.

Proof

Let and are two even numbers. So:

Put the two equations together side by side, and use the distributive property:

So we get the sum written in even number form.

#### Example2:

The sum of the two even numbers 12 and 16 is the even number 28:

### Note:

The sum of two odd numbers is an even number.

Proof

Let and are two odd numbers. So:

Put the two equations together side by side, and use the distributive property:

So we get the sum written in even number form.

#### Example3:

The sum of the two even numbers 11 and 19 is the even number 20:

### Note:

The sum of an odd number and an even number is an odd number.

Proof

Let an even number and an odd number. So:

Put the two equations together side by side, and use the distributive property:

So we get the sum written in odd number form.

#### Example4:

The sum of the even number 20 and the odd number 17 is the odd number 37:

### Note:

The product of two even numbers is an even number.

Proof

Let and are two even numbers. So:

Multiply the two equations side by side, and use the distributive property:

So we get the product written in even number form.

#### Example5:

The product of the two even numbers 12 and 4 is the even number 48:

### Note:

The product of two odd numbers is an odd number.

Proof

Let and are two odd numbers. So:

Multiply the two equations together side by side, and use the distributive property:

So we get the sum written in odd number form.

#### Example6:

The product of the two odd numbers 11 and 7 is the odd number 77:

### Note:

The product of an odd number and an even number is an even number.

Proof

Let an even number and an odd number. So:

Multiply the two equations together side by side, and use the distributive property:

So we get the sum written in even number form.

#### Example4:

The sum of the odd number 5 and the even number 10 is the even number 50:

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