Divisibility Rule of 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.

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.

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.

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.

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

• Divisibility Rule of 4
• Divisibility Rule of 5
• Divisibility Rule of 6
• Divisibility Rule of 7
• Divisibility Rule of 8
• Divisibility Rule of 9
• Divisibility Rule of 11
• Divisibility Rule of 13

What is the 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.