Divisibility Rules for 2, 3, 4, 5, 6, 7, 8, 9 & 10 | Handy Tricks

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Overview of Divisibility Rules:

What Are Divisibility Rules?

Divisibility rules are simple tests that help decide whether or not a number is divided by another number evenly and without performing the usual calculations.

Take a look at this scenario:

Aaron picks 568 apples from his orchard. He has only six cartons. Will Aaron be able to pack the apples evenly in the six cartons with no leftovers?

Apples picked



Cartons to pack

To solve a problem like this, ideally we would divide the number of apples by 6. But with divisibility rules at your fingertips, you can answer it easily without doing too much calculation.

In this lesson, we’ll walk you through the divisibility rules for numbers 2 through 10 and illustrate the concept with relevant examples.

Divisibility Rule for 2

All even numbers are divisible by 2. So, look for numbers that have 0, 2, 4, 6, or 8 as their last digit.

Examples

Is 286 divisible by 2?

Test

The number 286 has 6 as its last digit.

Result

286 is divisible by 2.

Is 3,527 divisible by 2?

Test

The last digit here, 7, is an odd number.

Result

3,527 is not divisible by 2.

Divisibility Rule for 3

Add up all the digits, and divide the sum by 3. If there are no remainders, the number will pass the divisibility test for 3.

Examples

Is 618 divisible by 3?

Test

The sum of the digits is:

6 + 1 + 8 = 15.

15 is a multiple of 3. (3 x 5 = 15)

Result

618 is divisible by 3.

Is 1,369 divisible by 3?

Test

The sum of the digits is:

1 + 3 + 6 + 9 = 19.

Dividing 19 by 3,

Dividing 19 by 3 - 1 left over

We have 1 left over.

Result

1,369 is not divisible by 3.

Divisibility Rule for 4

Pick the last two digits, and divide the number formed by 4. If there’s no remainder, the number is divisible by 4.

Examples

Is 778 divisible by 4?

Test

The number formed by the last two digits is 78.

Dividing 78 by 4,

Dividing 78 by 4 - 2 left over

We are left with 2 as the remainder.

Result

778 is not divisible by 4.

Is 12,456 divisible by 4?

Test

The number formed by the last two digits is 56.

Dividing 56 by 4,

Dividing 56 by 4 - no remainder

We have nothing left over.

Result

12,456 is divisible by 4.

Divisibility Rule for 5

A number is divisible by 5 if it has 0 or 5 in its ones place.

Examples

Is 154 divisible by 5?

Test

The number has 4 in the ones place.

Result

154 is not divisible by 5.

Is 66,505 divisible by 5?

Test

The number ends in 5.

Result

66,505 is divisible by 5.

Divisibility Rule for 6

If a number is divisible by 2 and 3, it is divisible by 6 as well.

A Venn diagram can really come handy to understand this rule.

venn diagram for divisibility by 6

As you can see the numbers 6, 12, and 18 in the intersection are divisible by both 2 and 3. Thus, they are divisible by 6 as well.

Examples

Is 7,284 divisible by 6?

Test

The last digit is 4.

⇨ 7,284 is divisible by 2.

The sum of the digits is:

7 + 2 + 8 + 4 = 21.

21 is a multiple of 3. (3 x 7 = 21)

⇨ 7,284 is divisible by 3.

Result

7,284 is divisible by 6.

Is 14,348 divisible by 6?

Test

The number is even with 8 as its last digit.

⇨ 14,348 is divisible by 2.

The sum of the digits is:

1 + 4 + 3 + 4 + 8 = 20.

Dividing 20 by 3,

Dividing 20 by 3 - 2 left over

We have 2 as the remainder.

⇨ 14,348 is not divisible by 3.

Result

14,348 is not divisible by 6.

Divisibility Rule for 7

Double the last digit and subtract it from the number formed by the other digits. If the difference is divisible by 7, the number is divisible by 7, too.

Examples

Is 644 divisible by 7?

Test

Doubling the last digit, we have 4 x 2 = 8.

The number formed by the other digits is 64.

Now, 64 – 8 = 56.

56 is a multiple of 7. ( 7 x 8 = 56)

Result

644 is divisible by 7.

Is 3,876 divisible by 7?

Test

Doubling the last digit, we have 6 x 2 = 12.

The number formed by other digits is 387.

Now, 387 – 12 = 375.

Let’s repeat the process.

Doubling the last digit 5 gives us 10.

The number formed by other digits is 37.

37 – 10 = 27.

Dividing 27 by 7,

Dividing 27 by 7 - 6 left over

We are left with 6 as the remainder.

Result

3,876 is not divisible by 7.

Divisibility Rule for 8

If the number formed by the last three digits is divisible by 8, the number is divisible by 8.

Examples

Is 28,256 divisible by 8?

Test

The number formed by the last three digits is 256.

Dividing 256 by 8,

Dividing 256 by 8 - no remainder

256 leaves no remainder when divided by 8.

Result

28,256 is divisible by 8.

Is 786,874 divisible by 8?

Test

The number formed by the last three digits is 874.

Dividing 874 by 8,

Dividing 874 by 8 - 2 left over

It leaves a remainder, 2.

Result

786,874 is not divisible by 8.

Divisibility Rule for 9

Add up all the digits and divide the sum by 9. If the sum is divisible by 9, the number is divisible by 9, too. Alternatively, you can repeat adding the digits until you end up with 9.

Examples

Is 65,547 divisible by 9?

Test

Adding all the digits, we have:

6 + 5 + 5 + 4 + 7 = 27.

27 is divisible by 9.

Also, adding the digits 2 and 7, we get 9.

Result

65,547 is divisible by 9.

Is 223,567 divisible by 9?

Test

The sum of the digits is:

2 + 2 + 3 + 5 + 6 + 7 = 25.

Repeating the process, we have 2 + 5 = 7 instead of 9.

Result

223,567 is not divisible by 9.

Note: The rule for 9 is similar to the rule for 3.

Divisibility Rule for 10

A number is divisible by 10 if it has zero as its last digit.

Examples

Is 89,340 divisible by 10?

Test

The number has 0 as the last digit.

Result

89,340 is divisible by 10.

Is 7,654 divisible by 10?

Test

The number ends in 4 and not zero.

Result

7,654 is not divisible by 10.

Lighten it Up with Some Number-Finding Fun

 Here’s a simple number trick that uses the divisibility rules.

 All you need to do is simply follow the instructions, and we’ll find the number on your mind.

 Think of any natural number.

 Double it and multiply by 5.

 Now, the number you had in your mind is simply the product without the last digit.

 Intrigued about the math that helped us amuse you? Click here!

   Here, you are asked to double the number and multiply it by 5, which makes it divisible by 2 and 5.

   When a number is divisible by 2 and 5, the product is divisible by 10.

   So we simply ignore the zero at the end of the product, and the number is unveiled.

   What are you waiting for? Play it on your friends and leave them awestruck!

Interesting Facts

  • A number divisible by 8 is divisible by 4 and 2. Similarly, a number divisible by 9 is divisible by 3.

  • Numbers that have two zeros at the end are divisible by 4. Numbers with three zeros at the end are divisible by 8.

  • A number divisible by 10 is divisible by 2 and 5.

  • The sum of 3 consecutive numbers is divisible by 3.

  • The number 2,520 is the smallest number that is divisible by 2, 3, 4, 5, 6, 7, 8, 9, and 10. Can’t quite believe it? Check for yourself!

The Lesson in a Nutshell

Divisors
Divisibility Rules
2
The last digit is even.
3
The sum of the digits is divisible by 3.
4
The number formed by the last 2 digits is divisible by 4.
5
The last digit is 0 or 5.
6
The number is even and divisible by 3.
7
Double the last digit and subtract it from the number made by the other digits.
The result must be divisible by 7.
8
The number formed by the last 3 digits is divisible by 8.
9
The sum of the digits is divisible by 9.
10
The number ends in 0.

Take a Divisibility-Rules Quiz Now!

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