What are divisible by 3?

The divisibility rule of 3 states that if the sum of the digits of a whole number is a multiple of 3, then the original number is also divisible by 3. With the help of the multiplication table of 3 or by using skip counting by 3 (starting at 0 and adding 3) it is easy to find whether a smaller number is divisible by 3 or not. However, for larger numbers, we can check if that number is completely divisible by 3 or not without doing the actual division.

What is the Divisibility Rule of 3?

A whole number is said to be divisible by 3 if the sum of all digits of that whole number is a multiple of 3 or exactly divisible by 3.

Divisibility Rule of 3 with Examples

The divisibility rule for 3 can be understood with the help of the following examples.

Example: Test the divisibility of the following numbers by 3.

a.) 1377

b.) 2130

c.) 3194

Solution:

a) In 1377, the sum of all the digits = 1 + 3 + 7 + 7 = 18. Since 18 is divisible by 3, it means 1377 is also divisible by 3. Here, 1377 ÷ 3 = 459, where 459 is the quotient and 0 is the remainder.

b) In 2130, the sum of all the digits = 2 + 1 + 3 + 0 = 6. Since 6 is divisible by 3, it means 2130 is also divisible by 3. Here, 2130 ÷ 3 = 710, where 710 is the quotient and 0 is the remainder.

c) In 3194, the sum of all the digits = 3 + 1 + 9 + 4 = 17. Since 17 is not divisible by 3, it means 3194 is not exactly divisible by 3. Here, 3194 ÷ 3 = 1064, where 1064 is the quotient and the remainder is 2.

Divisibility Rule of 3 for Large Numbers

The divisibility rule of 3 for large numbers states that if the sum of all digits of a large number is divisible by 3 or is a multiple of 3 then we can say that the large number is also divisible by 3.
Example:

a) 220077
Here, the sum of all the digits = 2 + 2 + 0 + 0 + 7 + 7 = 18. We know that 18 is divisible by 3 which means 220077 is also divisible by 3. This can be verified as follows. 220077 ÷ 3 = 73359, where 73359 is the quotient and 0 is the remainder.

b) 1121031
Here, the sum of all the digits = 1 + 1 + 2 + 1 + 0 + 3 + 1 = 9. We know that 9 is divisible by 3 which means 1121031 is also divisible by 3. This can be verified as follows. 1121031 ÷ 3 = 373677, where 373677 is the quotient and 0 is the remainder.

c) 3456194
Here, the sum of all the digits = 3 + 4 + 5 + 6 + 1 + 9 + 4 = 32. We know that 32 is not divisible by 3 which means 3456194 is not completely divisible by 3.

Divisibility Rule of 3 and 9

The divisibility rule of 3 and the divisibility rule of 9 are slightly similar. As we already discussed above that the divisibility rule or divisibility test of 3 states that if the sum of all digits of a number is divisible by 3 then the number is also divisible by 3. Just like the divisibility rule of 3, the divisibility rule of 9 states that the number is said to be divisible by 9 if the sum of all the digits of a number is divisible by 9.

For example, 52884 is divisible by 3 as the sum of all digits that is 5 + 2 + 8 + 8 + 4 = 27 is divisible by 3. Here, 52884 ÷ 3 = 17628, where 17628 is the quotient and the remainder is 0. Note that the sum of the digits of the number 27 is 2 + 7 = 9 is also divisible by 3. We can repeat this process so that we get the sum closer to 3 and find out whether the number is divisible by 3 or not.

Divisibility Test of 3 and 4

The divisibility test of 3 and the divisibility test of 4 are completely different. The divisibility test of 3 states that the number is divisible by 3 if the sum of all digits of a number is divisible by 3, whereas, the divisibility test of 4 states that the number is said to be divisible by 4 if the last two digits of the given number are zeros or the number formed by the last two digits, that is, the digit at tens place and ones place is divisible by 4.

For example, 1236 is divisible by 3 as the sum of all digits that is 1 + 2 + 3 + 6 = 12. We know that 12 is divisible by 3. Now, 1236 is divisible by 4 as the number formed by the last two digits, that is, 36 is divisible by 4. Therefore, 1236 is also divisible by 4. This can be verified as follows. 1236 ÷ 4 = 309, where 309 is the quotient and the remainder is 0.

☛ Related Topics

1. Example 1: For the following numbers, using the test of divisibility by 3, find out whether the numbers are divisible by 3 or not.

   a.) 66 b.) 97

   c.) 32

Solution:

a) In number 66, the sum of all the digits is 6 + 6 = 12, which is divisible by 3. Therefore, 66 is also divisible by 3. b) In number 97, the sum of all the digits is 9 + 7 = 16, which is not divisible by 3. Therefore, 97 is not divisible by 3.

c) In number 32, the sum of all the digits is 3 + 2 = 5, which is not divisible by 3. Therefore, 32 is not divisible by 3.

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FAQs on Divisibility Rule of 3

The divisibility rule of 3 states that a whole number is said to be divisible by 3 if the sum of all its digits is exactly divided by 3. Without performing division we can find out whether a number is divisible by 3 or not. For example, 45 is divisible by 3 because the sum of 45 is (4 + 5) = 9, which is divisible by 3. Hence, 45 is said to be divisible by 3 because it gives the quotient as 15 and the remainder as 0.

Using the Divisibility Rule of 3, Check if 120 is Divisible by 3.

First, we need to check if the sum of all the digits of the given number is divisible by 3 or not. The sum of the digits of 120 = 1+ 2 + 0 = 3. We know that 3 is divisible by 3. Thus, 120 is divisible by 3.

What is the Divisibility Rule of 3 and 4?

According to the divisibility rule of 3, a number is said to be divisible by 3 if the sum of all digits of that number is divisible by 3. For example, the number 495 is completely divisible by 3. The sum of all digits are 4 + 9 + 5 = 18 and 18 is divisible by 3. Thus, 495 is divisible by 3, where quotient = 165 and remainder = 0. Let us take another example, the number 55 is not exactly divisible by 3 as the sum of all digits of the number 55 is 5 + 5 = 10 and 10 cannot be completely divided by 3. If 55 is divided by 3 the quotient will come to 18 and the remainder will come to 1.

According to the divisibility rule of 4, if the number formed by the last two digits is divisible by 4 or the number has two zeros in the end then the number is divisible by 4. For example, 4420 is divisible by 4 as the number formed by the last two digits, that is, 20, is divisible by 4[20 ÷ 4 = 5].

How do you know if a Big Number is Divisible by 3?

According to the divisibility rule of 3, any big number is exactly divisible by 3 if the sum of the digits is a multiple of 3. For example, the number 2,146,497 is exactly divisible by 3, where quotient = 715,499 and remainder = 0. The sum of all digits is 2 + 1 + 4 + 6 + 4 + 9 + 7 = 33 and 33 is exactly divisible by 3.

Using the Divisibility Rule of 3, Check if 195 is Divisible by 3.

The divisibility rule of 3 states that if the sum of the digits of a given number is divisible by 3 then the number is also divisible by 3. So, the sum of the digits of 195 is (1 + 9 + 5) = 15, which is exactly divisible by 3. Thus, 195 is divisible by 3.

• The rule for divisibility by 3 works for all numbers no matter how large.
• Add the digits of the number and check if this result is also divisible by 3.
• We add the individual digits of the number 7, 749, 984.
• 7 + 7 + 4 + 9 + 9 + 8 + 4 = 48.
• We might not be sure if 48 is divisible by 3.
• The rule can be applied again to 48 to see if 48 is divisible by 3.
• 4 + 8 = 12, which is in the 3 times table.
• 12 is in the three times table, therefore so is 48 and so is 7, 749, 984.
• 7, 749, 984 is divisible by 3, which means that 7, 749, 984 is in the 3 times table.

To test if a number is divisible by 3, follow these steps:

1. Add the individual digits of the number to make a total.
2. If this total is divisible by 3, the original number is divisible by 3.
3. If you are not sure if the total is divisible by 3, apply the first two steps to that number.

If a number is divisible by 3, it means that the number is in the 3 times table. A number that is divisible by 3 is a multiple of 3. The number can be divided exactly by 3 to leave no remainder.

In this example, we will use the divisibility by 3 rule to test 5502. Is 5502 in the 3 times table?

A number is divisible by 3 if the sum of its digits is also divisible by 3. For example, 5502 is divisible by 3 because 5 + 5 + 0 + 2 = 12. 12 is divisible by 3 and so, 5502 is divisible by 3.

The first step is to add the digits of the number.

5 + 5 + 0 + 2 = 12.

The next step is to check if this new, smaller number is divisible by 3.

12 is 4 × 3. 12 is in the three times table and so, 5502 is also in the three times table.

We can also check that each number in the working out is a multiple of 3 by adding its digits.

12 is a multiple of 3 because 1 + 2 = 3.

Here is another example of using the rule for divisibility by 3 to test if a number is a multiple of 3.

Is the number 409 a multiple of 3?

The first step is to add the digits of the number.

4 + 0 + 9 = 13.

The next step is to decide if the sum of the digits is a multiple of 3.

13 is not a multiple of 3. It is not in the 3 times table. We can also see that 1 + 3 = 4 and 4 is not a multiple of 3, therefore 13 is not a multiple of 3.

13 is an example of a number that is not divisible by 3 and so, 409 is not divisible by 3.

A number is not divisible by 3 if the sum of its digits is not divisible by 3. 409 is not divisible by 3 because 4 + 0 + 9 = 13 and 13 is not divisible by 3.

Prime numbers are not divisible by 3 because they are ony divisible by 1 and themselves. For example, 13 is a prime number and so it not divisible by 3.

The rule for divisibility by 3 works for all numbers no matter how large.

For example, here is the number 529, 943.

The first step is to add the digits of the number.

5 + 2 + 9 + 9 + 4 + 3 = 32

The next step is to test if the sum of the digits is divisible by 3.

32 is not divisible by 3. We know this because 30 and 33 are multiples of 3 and 32 is in between these numbers.

We can also use the same test on 32 to show that it is not divisible by 3. We add the digits. 3 + 2 = 5 and 5 is not a multiple of 3.

32 is not a multiple of 3 and therefore 529, 943 is not a multiple of 3 either. 529, 943 is an example of a number that is not divisible by 3.

Here is an example of testing a large number to see if it is in the 3 times table.

Is 7, 749, 984 a multiple of 3?

Start by adding the digits.

7 + 7 + 4 + 9 + 9 + 8 + 4 = 48

48 is 16 × 3 and so, it is a multiple of 3.

If we were not sure if the number was a multiple of 3, add the digits and see if the total is a multiple of 3.

4 + 8 = 12, which is 4 × 3. 12 is a multiple of 3 and so, 48 is a multiple of 3 and 7, 749, 984 is also a multiple of 3

Why Does the Divisibility Rule for 3 Work?

The divisibility rule for 3 works because the number represented by each digit can be written as a multiple of 9 plus that digit. 9 is divisible by 3 so if the sum of the digits is divisible by 3, the number itself is too.

Here is the proof that 3174 is divisible by 3.

The digit 3 stands for 3000, which is 1000 × 3.

This is the same as 999 × 3 plus one more 3.

The digit 1 stands for 100, which is 100 × 1.

This is the same as 99 × 1 plus one more 1.

The digit 7 stands for 70, which is 10 × 7.

This is the same as 9 × 7 plus one more 7.

The digit 4 stands for one lot of 4.

The multiples of 9, 99 and 999 are all divisible by 3.

3174 is equal to 999 × 3 + 99 × 1 + 9 × 7 plus 3 + 1 + 7 + 4.

The multiples of 9, 99 and 999 are all divisible by 3, so we only need to check the sum of the digits: 3 + 1 + 7 + 4.

3 + 1 + 7 + 4 = 15, which is divisible by 3.

Since the multiples of 9, 99 and 999 along with the sum of the digits are all divisible by 3, the entire number is divisible by 3.

All numbers are written in base 10. This means that the digits of each number represent a multiple of 9 plus that digit. The multiples of 9 are divisible by 3, so we simply need to test the sum of the digits to see if the whole number is divisible by 3.

No matter what the number, add the digits to check if it is divisible by 3.

List of Numbers Divisible by 3

Here is a list of 2-digit numbers less than 100 that are divisible by 3:

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 and 99.

There are 33 2-digit numbers that are divisible by 3. The largest 2 digit number divisible by 3 is 99, which is 33 × 3.

This list can help us to identify some common multiples of 3, which helps us to identify if a larger number is divisible by 3.

Prime numbers are not be divisible by 3. This is because prime numbers can only be divided by 1 and themselves.

Even numbers can be divisible by 3. For example, the even number of 12 is divisible by 3.

All numbers that are divisible by 9 are also divisible by 3. This is because 3 divides exactly into 9.