Even Numbers
Overview
Even numbers are amounts that can be exactly divided by two. You can always create two equal groups out of an even number of objects!
Even numbers are easy to spot because they end in…
0, 2, 4, 6, or 8
Take a look at the number chart below. The even numbers are shaded in different colors.
When you count just the even numbers starting with zero, you are Skip-Counting by Two. How high can you count just by even numbers? You can use the number chart below to help you.
We use even numbers very often. For example, when counting big amounts of objects so that we can go fast by counting two at a time. Also, there are plenty of things around us that come in even numbers: a butterfly has 2 wings, a car has 4 wheels, we have 10 fingers, just to mention some.
But how are the even numbers defined? and which are their main properties? That’s what we are about to discover.
What are even numbers?
Intuitively, we think of an even number as a number that can be split into two equal parts. Let’s imaging a group of sea animals somewhere in the deep ocean… Just like in the picture!
Can we split the group of fishes into two subgroups of equal size? Yes, we can! As in the picture below. It means that 6 (the number of fishes) is an even number because it can be split into two groups of size 3 each.
What about the number of starfishes? Is it even? Well, if we split the group of 3 starfishes into two subgroups, we will always get one more star in one of the subgroups, as shown below. Thus, 3 is not an even number, because it cannot be split into two equal parts.
In other words, when we split the 3 starfishes into two parts, they seem “odd” because they don’t look harmonious or balanced. That is how we call the numbers that are not even: they are odd numbers.
By the way, 4 is also an even number! To convince ourselves, try to split the set of 4 sea horses into two equal parts: as below, each part has 2 sea horses.
Formally, we say that an integer number is even if, when divided by 2, we get 0 as a remainder. Or, equivalently, an integer number is even if 2 is one of its factors. Like in the ocean example: 6 = 2 × 3 meaning that when we divide 6 by 2, we get 3 as a quotient.
This quotient corresponds to the number of fishes in each equal part. Also, 4 = 2 × 2, then if we divide 4 into 2 parts, each part will have 2 elements.
We know that when an integer number is divided by 2, there are two possible remainders: 0 or 1. What happens when the remainder is 1? Well, in that case, we say that the number is odd. In other words, an integer number is odd if 2 is not one of its factors. Like what happened with the starfishes: 2 is not a factor of 3, thus 3 is odd, it is not even!
Other even numbers are, for example, 10, 36, and 568 because 10 = 2 × 5, 36 = 2 × 18, and 568 = 2 × 284. Can we see the pattern in these expressions? Let’s discuss that next.
How to recognize even numbers?
First of all, since any even number has 2 as one of its factors, any even number is of the form
where k is an integer.
Look at the examples before: 10 = 2 × 5, thus k = 5; and 36 = 2 × 18, thus k = 18 in this case. In fact, we get an even number for each value of k.
For example, if k = 25 then 50 = 2 × 25 is even. Also, the value of k can be negative; for example, if k= -7 then -14 = 2 × (-7) is an even number.
We can recognize even numbers in the number line as well, where they are located alternately. In the picture below, we can see the even numbers: -8, -6, -4, -2, 0, 2, 4, 6, 8 and 10.
Yes! 0 is an even number since it can be written as 0 = 2 × 0. Also, intuitively, if we start with 0 elements and we split them into two parts, each part will have an equal size: 0 elements.
We see, in the number line, that each even number has two odd numbers as neighbors on both sides.
For example, 4 has 3 and 5 on both sides, which are odd numbers. Likewise, -6 has -7 and -5 on both sides, which are odd numbers too.
The first even whole numbers are: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, and so on.
Notice in the number line, that between 6 and 8, for example, there isn’t any other even number. When this happens, the numbers are called consecutive even numbers. Similarly, 2 and 4, -6 and -4 are consecutive even numbers.
However, 4 and 8 are not consecutive because 6 is another even number between them. Any integer between two consecutive even numbers is an odd number. For example, 7 is between the consecutive even numbers 6 and 8.
What if we want to list the even numbers between 105 and 120? Well, since 2 × 52 = 104 and 2 × 53 = 106, then 104 and 106 are consecutive even numbers, and 105 is not even. Thus, the first number on the list is 106.
Precisely, the numbers in the list are of the form 2 × k, where k is greater than 52 and less than or equal to 60, as is shown in the table.
|
Value of k |
Value of 2×k |
|
53 |
106 |
|
54 |
108 |
|
55 |
110 |
|
56 |
112 |
|
57 |
114 |
|
58 |
116 |
|
59 |
118 |
|
60 |
120 |
Let’s now consider a big number like 3,187,074. Is it even? To determine if any given number is even, it is enough to look at the digit in the ones place. If the digit in the ones place of a number is 0, 2, 4, 6 or 8, then the number is even.
Thus, our big number 3,187,074 is even because the digit in the ones place is 4. Contrarily, 24,046,243 is not even because the digit in the ones place is 3, which is different from 0, 2, 4, 6, and 8.
Did you know that…?
Almost always corn have an even number of rows on each cob? In fact, corn with an odd number of rows are weirderthan four-leaf clovers! In average, mostcobs have 16 rows.
Let us practice what we have learned so far with the following example.
Practice Question 1
At a birthday party, there was an odd number of kids. Each kid ate the same even number of muffins. Among all the kids, 60 muffins were eaten.
How many kids were there at the party?
Practice Question 2
Let’s keep counting by even numbers!
What even numbers come next?
2, 4, 6, ___, ___, ___
Operations with even numbers
Let’s now explore what happens when we add, subtract, multiply or divide two even numbers. Is the result always even?
The sum of two even numbers is an even number: two even numbers are written as 2×k and 2×k’. Thus, their sum (2×k) + (2×k’)= 2×(k+k’) is also an even number.
For example, 26 and 12 are even because their digits in the ones place are 6 and 2, respectively. Their sum is 26 + 12 = 38, which is also even because 38 = 2 × 19.
The subtraction of two even numbers is an even number: two even numbers are written as 2×k and 2×k’. Thus, their subtraction (2×k) – (2×k’)= 2×(k-k’) is an even number.
If we subtract 38 and 4, we get 38-4=34. The digit in the ones place of 34 is 4, thus it is an even number.
The product of two even numbers is an even number: two even numbers are written as 2×k and 2×k’. Thus, their product (2×k) × (2×k’) = 2×(k×2×k’) is also an even number.
For example, 34 × 12 = 408 is also even, because its digit in the ones place is 8.
! The division of two even numbers is not necessarily an even number.
It is very important to notice that when dividing two even numbers, it is possible to get an odd numberas a result. For example, 4 and 12 are both even, but 12÷4 = 3 which is not even!
Practice Question 3
Two friends gather together to play with their favorite toys. Mark brought 8 cars, and Brian brought an even number of action figures.
Is the total number of toys they brought even or odd?
Practice Question 4
On a school trip, a group of students visited a theme park. Each of the students rode 4 roller coasters, 2 dark rides, 2 movie rides, and 6 flat rides. One of the students estimated that they rode an odd number of attractions in all.
Is that possible?
Practice Question 5
Look at the messy group of numbers below.
Can you spot the even numbers?
Practice Question 6
You are getting good at even numbers!
What even numbers are missing below?
Practice Question 7
Are all the numbers below even numbers? Check your answer!
12, 24, 36, 48, 60, 72, 84, 96, 100
Practice Question 8
Count the dinosaurs below. Is there an even number of dinosaurs?
FAQs
What is an even number?
It is a number that has 2 as one of its factors, and can therefore be split into two equal parts. They are written in the form 2×k. For example, 4, 16 or 38.
Do all even numbers have 0 or 2 in the ones place?
No, the digit in the ones place of an even number is 0, 2, 4, 6 or 8. For example, 56 is even and its digit in the ones place (6) is neither 0 nor 2.
Is it zero an even number?
Yes, zero is written as 0 = 2×0 which makes it an even number.
Operating two even numbers always gives an even number?
No, if we divide two even numbers, the result may be an odd number. For example, 20 and 4 are both even, but 20÷4 =5 is not even. However, if we add, subtract or multiply two even numbers, the result is always an even number.
How many even numbers there are?
There are infinitely many even numbers. Any number of the form 2×k is even, where k can be any integer number, thus we have infinitely many possible choices for k.
Which are the first even numbers?
Since there are infinitely many negative even numbers, we should wonder about the first whole even numbers. The first ten whole even numbers are: 0, 2, 4, 6, 8, 10, 12, 14, 16, and 18.