# Divisibility rules

This lesson presents divisibility rules for the numbers 2, 3, 4, 5, 6, 7, 8, 9, and 10. Divisibility rules of whole numbers are very useful because they help us to quickly determine if a number can be divided by 2, 3, 4, 5, 9, and 10 without doing long division. This is especially useful when the numbers are large.

In general, a whole number x divides another whole number y if and only if you can find a whole number n such that x × n = y.

For instance, 12 can be divided by 3 because 3 × 4 = 12 Divisibility means that you are able to divide a number evenly. For instance, 8 can be divided evenly by 4 because 8/4 = 2. However, 8 cannot be divided evenly by 3.

To illustrate the concept, let's say you have a cake and your cake has 8 slices, you can share that cake between you and 3 more people evenly. Each person will get 2 slices.

However, if you are trying to share those 8 slices between you and 2 more people, there is no way you can do this evenly. One person will end up with less cake.

## Divisibility rules and examples showing how to use the rules

Rule #1: divisibility by 2

A number is divisible by 2 if its last digit is even or the last digit is 0,2,4,6,or 8.

For instance, 8596742 is divisible by 2 because the last digit is 2.

Rule # 2: divisibility by 3:

A number is divisible by 3 if the sum of its digits is divisible by 3.

For instance, 3141 is divisible by 3 because 3+1+4+1 = 9 and 9 is divisible by 3.

Rule # 3: divisibility by 4

A number is divisible by 4 if the number represented by its last two digits is divisible by 4.

For instance, 8920 is divisible by 4 because 20 is divisible by 4.

Rule #4: divisibility by 5

A number is divisible by 5 if its last digit is 0 or 5.

For instance, 9564655 is divisible by 5 because the last digit is 5.

Rule # 5: divisibility by 6

A number is divisible by 6 if it is divisible by 2 and 3. Be careful! it is not one or the other. The number must be divisible by both 2 and 3 before you can conclude that it is divisible by 6.

Rule # 6: divisibility by 7

To check divisibility by 7, study carefully the following two examples:

Is 348 divisible by 7?

Remove the last digit, which is 8. The number becomes 34. Then, Double 8 to get 16 and subtract 16 from 34.

34 − 16 = 18 and 18 is not divisible by 7. Therefore, 348 is not divisible by 7

Is 37961 divisible by 7?

Remove the last digit, which is 1. The number becomes 3796. Then, Double 1 to get 2 and subtract 2 from 3796.

3796 − 2 = 3794, and 3794 is still too big. Thus repeat the process.

Remove the last digit, which is 4. The number becomes 379. Then, Double 4 to get 8 and subtract 8 from 379.

379 − 8 = 371,  and 371 is still too big. Thus repeat the process.

Remove the last digit, which is 1. The number becomes 37. Then, Double 1 to get 2 and subtract 2 from 37.

37 − 2 = 35 and 35 is divisible by 7. Therefore, 37961 is divisible by 7.

Rule #7: divisibility by 8

A number is divisible by 8 if the number represented by its last three digits is divisible by 8.

For instance, 587320 is divisible by 8 because 320 is divisible by 8.

Rule #8: divisibility by 9

A number is divisible by 9 if the sum of its digits is divisible by 9.

For instance, 3141 is divisible by 9 because 3+1+4+1 = 9 and 9 is divisible by 9.

Rule # 9: divisibility by 10

A number is divisible by 10 if its last digit or the digit in the ones place is 0.

For instance, 522480 is divisible by 10 because the last digit is 0.

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