In mathematics, there is a wide array of techniques and concepts to learn. Of course, you have basic addition, subtraction, multiplication, and division, but all of these have their own methods and concepts to learn.
For instance, the act of dividing involves using various methods, learning how to divide fractions and decimals, understanding how to convert fractions to decimals, and much more. However, one of the more basic concepts is divisibility.
Divisibility enables us to understand the relationship between integers (whole numbers). Perhaps the most important number in divisibility is 2, as it has a wide variety of use cases, both in academia and real life.
To help you understand what it is, this article will explore the concept of divisibility, particularly as it pertains to the digit 2, by looking at what numbers are divisible by it, what numbers aren’t, and a divisibility rule you can apply to any number you come across.
Any even number is divisible by 2. This is by definition since all even numbers can be divided by 2 without leaving a remainder. This means that any number ending in 2, 4, 6, 8, or 0 satisfies this criterion and is, thus, divisible by 2.
As such, a nifty little shortcut you can use to determine whether a number is divisible by 2 is to ignore all the digits of the number except the last. If the last digit is even, it means the entire number is divisible by 2. What we’ve just described is called a 'divisibility rule' – something we’ll expand on later in this article. But before we get into that, let’s explore which numbers are not divisible by 2.
All odd numbers are not divisible by 2. This is by definition since an odd number is defined as a number that cannot be divided by 2 without leaving a remainder. This means you won’t be able to divide a number ending in 1, 3, 5, 7, or 9 by 2.
There are a lot of terms when it comes to division, and they all sound quite similar, such as dividend, divisor, and divisible. To avoid confusion, let’s take a step back and explain what divisible means.
Suppose a number is divisible by another number. It means it can be split into equal parts without leaving a remainder, meaning the final answer is an integer. In other words, a given number can be divided equally by another number.
For example, suppose you want to solve the following problem: 10 ÷ 2. The number 10 is divisible 2 because 10 divided by 2 equals 5 (10 ÷ 2 = 5) – 5 is a whole number, and there are no remainders left over. You can use this principle to see if any number is divisible by another number.
Suppose you want to solve 24 ÷ 4. The number 24 is divisible by 4 because 24 divided by 4 equals 6 (24 ÷ 4 = 6) – 6 is a whole number with no remainders.
Conversely, say you want to solve 10 ÷ 3. 10 is not divisible by 3 because 10 divided by 3 equals 3.33 (10 ÷ 3 = 3.33). Since 3.33 is not a whole number and has a remainder, the equation does not satisfy the criteria of being a number that can be divided by another number without a remainder.
It’s important to understand what it means because there are a few words that sound similar but have very different meanings. For instance, you have a word ‘dividend’, ‘divisor’, and even division itself.
Now that we’ve cleared that out of the way, let’s look at how you can determine whether a number is divisible by another in a matter of seconds.
They are a set of rules that can help you determine whether a number is or is not divisible by another without having to solve for the entire equation. This is particularly helpful if you encounter large numbers where solving the problem can take time.
There are different divisibility rules for each digit you are dividing by, and they can go up to infinity. But, instead of explaining all rules for all numbers – we’d be here all day if we did – in this section, we’ll focus on the divisibility rules for the digits 1 to 10.
Divisibility rule for 1
All whole numbers are divisible by 1, regardless of whether it’s a small number like 9 or a large number like 3,497.
Divisibility rule for 2
All even numbers are divisible by the number 2. Simply look at the last digit of any number; if it ends in 2, 4, 6, 8, or 0, it will be divisible by 2.
Divisibility rule for 3
If the sum of the digits is divisible by 3, then the original number is also divisible by 3.
To do this divisibility test, first, add all the digits of the original number together, and keep doing this until you have one digit remaining. All that remains is to check whether the remaining digit is divisible by 3.
For example, suppose you have the following equation: 288 ÷ 3.
- Add each digit together until there’s one remaining: 2 + 8 + 8 = 18
- Add each digit together until there’s one remaining: 1 + 8 = 9
- 9 ÷ 3 = 3
Since 9 was divisible by 3, 288 is also divisible by 3.
Divisibility rule for 4
If the last two digits are divisible by 4, then the original number is also divisible by 4.
For instance, suppose you have the following equation: 952 ÷ 4.
- Take the last two digits and divide them by 4: 52 ÷ 4 = 13
Since 52 is divisible by 4, 952 is also divisible by 4.
Divisibility rule for 5
If the last digit ends in 0 or 5, then the original number is also divisible by 4.
For example, suppose you have the numbers 35 and 780. Are these numbers divisible by 5? Since 35 ends in 5 and 780 ends in 0, both numbers are divisible by 5.
Divisibility rule for 6
If the number can be divided by both 2 and 3, then it is also divisible by 6.
Take the example of 24. 24 is both divisible by 2 (24 ÷ 2 = 12) and 3 (24 ÷ 3 = 8). Therefore, it’s also divisible by 6 (24 ÷ 6 = 4).
Divisibility rule for 7
If the result of multiplying the last digit and subtracting it from the number created by the other digits is divisible by 7, the original number is also divisible by 7. Sounds confusing, right? Let’s look at an example.
Take the equation 133 ÷ 7.
- Multiply the last digit by 2: 3 x 2 = 6
- Subtract it from the other digits: 13 - 6 = 7
Since 7 is divisible by 7 (7 ÷ 7 = 1), 133 is also divisible by 7.
Divisibility rule for 8
If the result of adding the last digit to 2 times the other digits is divisible by 8, then the original number is also divisible by 8. This is another one that can seem confusing but will make sense once shown.
Suppose you have the equation 64 ÷ 8.
- Multiply all digits except the last by 2: 6 x 2 = 12
- Add the last digit: 12 + 4 = 16
Since 16 is divisible by 8 (16 ÷ 8 = 2), 64 is also divisible by 8.
Divisibility rule for 9
If the sum of the digits is divisible by 9, then the original number is also divisible by 9. This works exactly the same as the divisibility rule for 3.
Say you have the equation 441 ÷ 9.
- Add each digit together: 4 + 4 + 1 = 9
Since 9 is divisible by 9 (9 ÷ 9 = 1), 441 is also divisible by 9.
Divisibility rule for 10
If the number's last digit ends in 0, it is divisible by 10. This is a fairly easy one, but let’s see an example.
Take the equation 320 ÷ 10. Since the last digit is 0, it means that 320 is divisible by 10.
Example 1 – Is the following number divisible by 2: 3,676
To determine whether a number is divisible by 2, you must see whether it is an even number. The last digit of an even number must be either 2, 4, 6, 8, or 0. The last digit of 3,676 is 6, showing an even number.
Therefore, 3676 is divisible by 2.
Example 2 – Is 285 divisible by 2?
285 is not an even number since the last digit, 5, is odd. As we’ve established, odd numbers are not divisible by 2.
Therefore, you can conclude that 285 is not divisible by 2.
Example 3 – Are the following numbers divisible by 2: 72 & 9,118
The last digit of 72 is 2, and the last digit of 9,118 is 8. Both 2 and 8 are even, meaning that both numbers are even.
Therefore, both 72 and 9118 are divisible by 2.
Example 4: Are the following numbers divisible by 2: 26, 194, & 735
The last digit of 26 is 6, which is an even number. The last digit of 194 is 4, which is also an even number. However, the last digit of 735 is 5, which is an odd number.
Therefore, you can conclude that 26 and 194 are divisible by 2, but 735 is not.
It helps in simplifying mathematical operations such as fractions or equations. For instance, the fraction 6/10 can be easily simplified since both numbers are divisible by 2. 6 becomes 3, and 10 becomes 5 to create the simplified fraction 3/5.
It also helps to find the highest common factor of a number. Additionally, it's used heavily in Number Theory, a field of mathematics that deals with the properties and relationships between integers. This has uses in computer science, programming, cryptography, and more.
Therefore, learning which numbers are divisible by 2 – and other numbers for that matter – has important academic and real-life applications.
Any even number is divisible by 2. A quick check to see if a number is even or odd is to look at the last digit. If it ends in 2, 4, 6, 8, or 0, it is an even number and can therefore be divisible by 2. This check is known as the divisibility rule for 2 and can be applied to any integer, big or small.
Whilst divisibility rules can seem rudimentary and insignificant, it plays a vital role in mathematics and everyday life, making them essential to learn.