The Divisibility Rules: 3, 6, 9

Have you ever wondered why some numbers will divide evenly (without a remainder) into a number, while others will not? The Divisibility Rules help us to determine if a number will divide into another number without actually having to divide. There is a divisibility rule for every number. However, some of the rules are easier to use than others. For the rest, it might just be simpler to actually divide.

Here is a look at the rules for 3, 6, and 9.
The Rule for 3: A number is divisible by 3 if the sum of the digits is divisible by 3.

What does this mean? This means that we need to add up the digits in the number and see of the answer is can be divided by 3 without a remainder.



Example: 34,911

Step 1: Add up the digits. 3 + 4 + 9 + 1 + 1 = 18

Step 2: Determine if 3 divides evenly into the sum of 18. Yes, 3 x 6 = 18. So 3 goes evenly into 18.

Step 3: Use the result to determine if 3 goes into 34,911. Because 3 divides into 18 evenly, 3 also divides evenly into 34,911. So 3 is a factor of 34,911.



Example: 45,799

Step 1: 4 + 5 + 7 + 9 + 9 = 34

Step 2: 3 does not divide evenly into 34.

Step 3: Therefore, 3 also does not divide evenly into 45,799. So 3 is not a factor of 45,799.

The Rule for 6: The prime factors of 6 are 2 and 3. So for a number to be divisible by 6, it must also be divisible by 2 and 3. Therefore, we need to check if a number is even and then check if the sum of the digits is divisible by 3.



Example:23,908

Step 1: Determine if the number is even. It ends in 8, so this number is even. Therefore it is divisible by 2.

Step 2: Add the digits together. 2 + 3 + 9 + 0 + 8 = 22

Step 3: 3 does not divide evenly into 22. So this number is not divisible by 3.

Step 4: Because this number is only divisible by 2, and not by 3, it is NOT divisible by 6.



Example:154,608

Step 1: This number is even and is therefore divisible by 2.

Step 2: 1 + 5 + 4 + 6 + 0 + 8 =24

Step 3: 24 is divisible by 3 because 3 x 8 = 24.

Step 4: Because the number is divisible by 2 and 3, it is also divisible by 6.

The Rule for 9: The prime factors of 9 are 3 and 3. So we can use a very similar rule to determine if a number is divisible by 9. Basically, we will see if the sum of the digits is divisible by 9. If it is, then the actual number is also divisible by 9. This is done the same way we checked the rule for 3.



Example: 871,989

Step 1: Add up the digits. 8 + 7 + 1 + 9 + 8 + 9 = 42.

Step 2: Determine if 9 goes into 42 evenly. We know that 9 x 4 = 36 and 9 x 5 = 45. So 9 will not go into 42 evenly.

Step 3: 871,989 is not divisible by 9.



Example: 92,745

Step 1: 9 + 2 + 7 + 4 + 5 = 27

Step 2: 9 goes into 27 3 times because 9 x 3 = 27.

Step 3: Because 9 goes into 27 evenly, it also goes into 92,745 evenly. Therefore, 92,745 is divisible by 9.

Using the divisibility tests, we can easily determine if a number is divisible by 3, 6 or 9. This will help you to also determine if a number is prime or not. When a number is divisible by something other than 1 or itself, it is not prime. So, if we can show that 3, 6 or 9 go into the number, we know that the number is not prime without doing any long division.


Related Links:
Math
Division
Division Worksheets
Divisibility Rules Worksheets
Divisibility Rules Quiz
Divisibility Test
Fractions
Factors


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