Even and odd numbers

Even and odd numbers are straightforward concepts. I will start easy, but I will try to challenge the topic a little bit.

What are even numbers?

An even number is any number that gives a remainder of zero when divided by 2. For example, 12 gives a remainder of 0 when divided by 2, so 12 is even.

You can also say that 12 is even because there exist a number 6, such that 12 = 2 × 6. Having said that, we can come up with a formal definition.

Formal definition of an even number:

A number n is even if there exist a number k, such that n = 2k  where k is an integer.

This is the formal way of saying that if n is divided by 2, we always get a quotient k with no remainder. Having no remainder means that n can in fact be divided by 2. 

How to quickly check if a number is even?

We saw in divisibility rules that a number is divisible by 2 or gives a remainder of zero if its last digit is 0,2,4,6,or 8.

Therefore, any number whose last digit is 0, 2, 4, 6, or 8 is an even number. For example, 98716 is an even number because its last digit shown in red is 6.

Other examples of even numbers are 58, 44884, 998632, 98, 48, and 10000000.

The figure below shows the difference between even numbers and odd numbers.

Even and odd numbers

What are odd numbers?

An odd number is any number that gives a remainder of 1 when divided by 2. For example, 27 gives a remainder of 1 when divided by 2, so 27 is odd.

You can also say that 27 is odd because there exist a number 13, such that 27 = 2 × 13 + 1. Having said that, we can come up with a formal definition.

Formal definition of an odd number:

A number n is odd if there exist a number k, such that n = 2k + 1  where k is an integer.

This is the formal way of saying that if n is divided by 2, we always get a quotient k with a remainder of 1. Having a remainder of 1 means that n cannot in fact be divided by 2.

How to quickly check if a number is odd?

Again, we saw in divisibility rules that a number is divisible by 2 or gives a remainder of zero if its last digit is 0,2,4,6,or 8.

Therefore, any number whose last digit is not 0, 2, 4, 6, or 8 is an odd number.

Other examples of odd numbers are 53, 881, 238637, 99, 45, and 100000023

Basic operations with even and odd numbers

Adding two even numbers

even + even = even

4 + 2 = 6

Generally speaking, let n1 = 2k1 be an even number and let n2 = 2k2 be another even number.

n1 + n2 = 2k1 + 2k2

n1 + n2 = 2(k1 + k2)

Notice that any number multiplied by 2 is even. This is the reason that 2(k1 + k2) is an even number!

Adding an even number to an odd number

even + odd = odd

6 + 3 = 9

Generally speaking, let n1 = 2k1 be an even number and let n2 = 2k2 + 1 be an odd number.

n1 + n2 = 2k1 + 2k2 + 1

n1 + n2 = 2(k1 + k2) + 1

Adding two odd numbers

odd + odd = even

13 + 13 = 26

Generally speaking, let n1 = 2k1 + 1 be an odd number and let n2 = 2k2 + 1 be another odd number.

n1 + n2 = 2k1 + 1 + 2k2 + 1

n1 + n2 = 2(k1 + k2) + 2

Notice that 2(k1 + k2) is even  and 2 is even.

Therefore, 2(k1 + k2) + 2 is even since adding two even numbers is equal to an even number

Multiplication

even × even = even

2 × 6 = 12

even × odd = even

8 × 3 = 24

odd × odd = odd

3 × 5 = 15

Subtraction

even − even = even

8 − 4 = 4

even − odd = odd

6 − 3 = 3

odd − odd = even

13 − 3 = 10

If you have any questions about this even and odd numbers lesson, just contact me.

Recent Articles

  1. Types of Triangles

    Dec 03, 22 12:47 PM

    Learn to identify 6 types of triangles based on their sides or angles and 7 more types based on both their sides and angles.

    Read More

  2. Box and Whiskers Plot

    Nov 18, 22 08:20 AM

    Easily learn to construct a box and whiskers plot for a set of data by using the median and the extreme values.

    Read More

Tough algebra word problems

100 Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

Math quizzes

 Recommended

Math vocabulary quizzes