# [Full Details] What are Even and Odd Numbers?

Numbers are not only a crucial part of mathematics but also play a vital role in our everyday life to count things. Numbers (more specifically integers) are classified into even and odd numbers. In this section, we will learn them together. If you are interested to know them separately then visit our pages below:

#### What are Even and Odd Numbers?

If a number is completely divisible by 2 and leaves a remainder 0, then that number is called an even number. For example, 2, 4, 10, etc are even numbers.

If a number is not divisible by 2 and leaves a remainder 1, then that number is called an odd number. For example, 1, 3, 7, 25, etc are odd numbers.

Remark: Note that there are infinitely many even numbers as well as odd numbers. The collection of all even numbers are …-6, -4, -2, 0, 2, 4, 6, … and the collection of all odd numbers are …-5, -3, -1, 3, 5, 7, …

#### Can Even and Odd Numbers be Negative?

Yes, even and odd numbers can be negative. Note that -2,-4, -6, -8 … are examples of negative even numbers, and -1, -3, -5, -7 … are examples of negative odd numbers.

Note that 0 is an even number and 1 is an odd number.

#### List of Even & Odd Numbers from 1 to 50

There are 25 even numbers and 25 odd numbers from 1 to 50. The even and odd numbers between 1 and 50 are given below.

 Even Numbers Odd Numbers 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49

#### General forms of Even and Odd Numbers & Set Representations

As even numbers are divisible by 2, we can write an even number as 2n for some integer n. On the other hand, since odd numbers leave a remainder 1 while dividing by 2, it can be expressed as 2n+1 where n is an integer. As a set, they can be respectively represented as

{2n: n is an integer}

and

{2n+1: n is an integer}.

Thus $2\mathbb{Z}$ and $2\mathbb{Z}+1$ respectively represent the set of all even and odd numbers. Here $\mathbb{Z}=$ {n: n is an integer} is the set of all integers.

#### Consecutive even and odd numbers

Two even numbers are said to be consecutive even numbers if their difference is 2. For example, 6 and 8 are consecutive even numbers as their difference is 8-6=2. But 4 and 8 are not as their difference not equal to 2.

Similarly, two odd are said to be consecutive if their difference is 2. Note that 1 and 3 are consecutive odd numbers.

Remark 1: The set of all even numbers {…-6, -4, -2, 0, 2, 4, 6, …} forms an arithmetic progression with a common difference 2. On the other hand, the set of all odd numbers {…-5, -3, -1, 3, 5, 7, …} also forms an arithmetic progression with a common difference 2.

Remark 2: The sum of first n even numbers is n(n+1). For a proof, visit 2+4+6+…+2n=n(n+1).

Remark 3: The sum of first n odd numbers is n2. For a proof, visit 1+3+5+…+(2n-1)=n2.

#### How to check a given number is even or odd?

To check a given number even or odd, we need to go through the following steps.

Step 1: At first, look at the given number and its last digit.

Step 2: If the number has the last digit either 2, 4, 6, 8, or 0 then the given number is an even number.

Step 2: If the number has the last digit either 1, 3, 5, 7, or 9 then the given number is an odd number.

For example, the number 24248 has the last digit 8, so it is an even number. As the number 76721 has unit digit 1, it is an odd number.

#### Properties of Even and Odd Numbers

1. The sum of an even number and an odd number is an even number. That is, Even + Odd = Even.

Proof: Let a be an even number and b be an odd number. By the definition of even and odd numbers, we can write

a=2m  and  b=2n+1  for some integers m and n.

a+b = 2m + 2n+1 = 2(m+n)+1

So we can write a+b = 2t+1 where t=m+n is an integer. This shows that a+b is an odd number. Hence we conclude that the sum of even and odd numbers is odd.

2. The product of an even number and an odd number is an even number. That is, Even × Odd = Even.

Proof: Let a be an even number and b be an odd number. As above we can write

a=2m  and  b=2n+1  for some integers m and n. Now multiplying we get

a×b = 2m(2n+1) = 4mn+2m = 2(2mn+m)

So we obtain that ab=2t where t=2mn+m is an integer. Thus ab is an even number, proving that the product of even and odd numbers is even.

3. The sum/difference of two even numbers is even. For more details, see even ± even = even.

4. The sum/difference of two odd numbers is even. For more details, visit odd ± odd = even.

5. The product of even numbers is again an even number. For a proof, see our page even × even = even.

6. The product of odd numbers is an odd number. For a proof, visit our page odd × odd = odd.

#### Key things to remember

• Even numbers are divisible by 2 but odd numbers are not.
• Even numbers leave remainder 0 and odd numbers leave remainder 1 while dividing by 2.
• Even number ends with 0, 2, 4, 6, or 8 whereas odd number ends with 1, 3, 5, 7, or 9.
• Even ± Even = Even
• Odd ± Odd = Even
• Even ± Odd = Odd
• Even × Even = Even
• Even × Odd = Even
• Odd × Odd = Odd

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