Have you ever wondered whether a particular number is divisible by 3 or not? Perhaps you’re cooking a meal for three people and need to know whether the portion size can be split evenly, or maybe you need to split the bill between three people.
Regardless of the reason, it can be helpful to know whether a number is divisible by 3 without taking out a pen and paper and doing a long-winded division calculation.
Fortunately, divisibility rules – also called divisibility tests – can provide you with the answer you’re looking for. But you might be asking, what is it?
We’ll explain what numbers are divisible by 3, the divisibility rule of 3, and plenty of examples to walk you through it.
All multiples of 3 are divisible by 3 since they can be divided into an integer (whole number) without leaving a remainder. For example, 6 is divisible by 3 because 6 ÷ 3 = 2. The answer (2) is a whole number and has no remainders.
Therefore, any multiple of 3 is considered to be divisible by 3. This also applies to both positive and negative numbers of all sizes.
When you divide zero by 3, the answer is zero, an integer with no remainder. Therefore, by definition of divisibility, zero is divisible by 3 – in fact, zero is divisible by all integers.
All multiples of 3, which include the following numbers:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99 etc.
We’ve only outlined the numbers up to 100, but this can continue until infinity. Remembering all those numbers will be nearly impossible without a photographic memory. Luckily, there’s a shortcut to determine which numbers are divisible by 3 without having to memorize them all and without having to solve the equation – the divisibility rule of 3.
The divisibility rule of 3 states that if the sum of the digits can be divided by 3, then the original number itself can also be divided by 3. This means that you must add all the digits of the number together, and if your resulting value is evenly divisible by 3, then so is the original number. Let’s look at an example to see how the method works.
Suppose you have the number 21. Is this number completely divisible by 3 using its divisibility rule?
First, you want to add all the individual digits of the given number (21) together:
- Sum of the individual digits: 2 + 1 = 3
The next step involves checking whether the answer (3) is divisible by 3. This should be obvious, but let’s follow the procedure just to see it in action:
- 3 ÷ 3 = 1
Our answer is 1, which is a whole number with no remainders. This means that 3 is divisible by 3. Knowing this, you can also conclude that 21 is also divisible by 3.
And that’s all there is to it. Learning the divisibility rule of 3 is incredibly easy and will only take a few seconds to calculate.
Now that you know the divisibility rule of 3, here are more examples of it in action.
Example 1 – two and three-digit numbers
Are the following numbers divisible by 3: 84, 27, 135, and 167.
The number 84 is divisible by 3 because:
- 8 + 4 = 12
- 12 ÷ 3 = 4, meaning 12 is divisible by 3 into an integer.
The number 27 is divisible by 3 because:
- 2 + 7 = 9
- 9 ÷ 3 = 3, meaning 9 is divisible by 3 into an integer.
The number 135 is divisible by 3 because:
- 1 + 3 + 5 = 9
- 9 ÷ 3 = 3, meaning 9 is divisible by 3 into an integer.
The number 167 is not divisible by 3 because:
- 1 + 6 + 7 = 14
- 14 ÷ 3 = 4 remainder 2, meaning 14 is not divisible by 3 into an integer.
Working with two or three-digit numbers is fairly easy, but how does it look when you have a much larger number?
Example 2 – four-digit numbers
Is the following number divisible by 3: 7,218, 6155, 2,096, and 3,342
The number 7,218 is divisible by 3 because:
- 7 + 2 + 1 + 8 = 18
- 18 ÷ 3 = 6, meaning 18 is divisible by 3 into an integer.
The number 6,155 is not divisible by 3 because:
- 6 + 1 + 5 + 5 = 17
- 17 ÷ 3 = 5 remainder 2, meaning 17 is not divisible by 3 into an integer.
The number 2,076 is divisible by 3 because:
- 2 + 0 + 7 + 6 = 15
- 15 ÷ 3 = 5, meaning 15 is divisible by 3 into an integer.
The number 3,342 is divisible by 3 because:
- 3 + 3 + 4 + 2 = 12
- 12 ÷ 3 = 4, meaning 12 is divisible by 3 into an integer.
Example 3 – five-digit numbers
Are the following numbers divisible by 3: 11,676, 46,139, 32,900, and 72,481.
The number 11,676 is divisible by 3 because:
- 1 + 1 + 6 + 7 + 6 = 21
- 21 ÷ 3 = 7, meaning 21 is divisible by 3 into an integer.
The number 46,139 is not divisible by 3 because:
- 4 + 6 + 1 + 3 + 9 = 23
- 23 ÷ 3 = 7 remainder 2, meaning 23 is not divisible by 3 into an integer.
The number 32,900 is not divisible by 3 because:
- 3 + 2 + 9 + 0 + 0 = 14
- 14 ÷ 3 = 4 remainder 2, meaning 14 is not divisible by 3 into an integer.
The number 72,483 is divisible by 3 because:
- 7 + 2 + 4 + 8 + 3 = 24
- 24 ÷ 3 = 8, meaning 24 is divisible by 3 into an integer.
As you can see, the divisibility rule of 3 can be tested on any number, regardless of its size. These examples show you its application on two, three, four, and five-digit numbers. However, the same principle can also be applied to six-digit or higher numbers.
Any number divisible by 9 is also divisible by 3. Additionally, the divisibility rule of 9 follows the same principles as the rule of 3, where you must add the sum of the digits together. From there, instead of dividing the resulting total by 3, it must be divided by 9. Therefore, if you learn how to perform the divisibility rule of 3, you’ll also learn how to perform the divisibility rule of 9.
For example, take the number 126. Let's use the divisibility rule of 9:
- 1 + 2 + 6 = 9
- 9 ÷ 9 = 1, meaning 126 is divisible by 9.
Now let's take the same number and use the divisibility rule of 3:
- 1 + 2 + 6 = 9
- 9 ÷ 3 = 3, meaning 126 is also divisible by 3.
As you can see, both divisibility rules follow the same process, and since 126 was divisible by 9, it was also found to be divisible by 3.
What does divisible mean?
Divisible means that a dividend can be divided by a divisor equally, leaving an integer (whole number) and no remainders. In other words, a number can be completely divided by another. For example, 6 is divisible by 2 since 6 ÷ 2 = 3.
What are divisibility rules?
A divisibility rule, also known as a divisibility test, is a quick test to determine whether a number is divisible by another without having to go through the actual division calculation.
What does it mean for a number to be divisible by 3?
It means that a number can be divided evenly by 3 with no remainder.
Are prime numbers divisible by 3?
A prime number is not divisible by 3. For example, 7 is a prime number and it cannot be divided by 3. However, 3 itself is a prime number and is, therefore, the only prime number divisible by 3.
Are all numbers ending in 3 divisible by 3?
Not all numbers ending in 3 are divisible by 3. It’s a common mistake that many beginners make, and the same applies to numbers ending in 6 and 9. Although these digits are multiples of 3, it doesn’t mean the entire number is divisible by 3. For instance, 16, 19, and 13 are not divisible by 3.
Can the divisibility rule of 3 be used for numbers greater than 100?
Absolutely, the divisibility rule of 3 can be applied to any integer, regardless of how big or small it is.
Any number that is a multiple of 3 is also divisible by 3. This may be easy to realize for small numbers but can get quite complicated for larger ones. To combat this, you can use the divisibility rule of 3 to help you identify which numbers are divisible by 3 and which aren't.
The divisibility rule of 3 states that if the sum of all the digits of the number is divisible by 3, then so is the original number. It’s easy as that, and it is a valuable tool to have in your mathematical repertoire.
