Session 5: Divisibility Rules for 2, 3, 4, 5, 6, 8, 9, 10 and 11 – Properties, Chart and Examples

So far, the only method we know to test whether or not a number is divisible by another number is to perform actual division and see whether or not the remainder is zero. This is tedious and time-consuming. However, there are some rules we can use to see if a large number is divisible by another number. 

What are Divisibility Rules?

Divisibility rules are shortcuts to determine if a number is divisible by another number without performing the actual division, usually by examining the digits of the given number. 

Martin Gardner, American mathematics and science writer, in his September 1962 “Mathematical Games” column in Scientific American article discussed divisibility rules for 2–12, where he explains that the rules were widely known during the 15th and 16th centuries in Renaissance Europe and used to reduce large number fractions to their lowest terms. 

In this section, we shall learn about some basic divisibility rules.

Divisibility Rule for 10

A natural number is divisible by 10 if its unit’s digit is 0.

Example of divisibility rule for 10:

The numbers 102030 and 98760 are divisible by 10 as they have 0 in their units’ place.

Divisibility Rule for 5

A natural number is divisible by 5 if its unit’s digit is either 0 or 5.

Example of divisibility rule for 5: 

The numbers 20400 and 8765 are divisible by 5 as they have 0 and 5 in their respective units’ place.

Divisibility Rule for 2

A natural number is divisible by 2, if its units digit is divisible by 2, i.e., the units place is either 0, 2, 4, 6 or 8.

Example of divisibility rule for 2:

The numbers 4546, 53952, 648394 are divisible by 2 as they end with 6, 2 and 4, respectively.

Divisibility Rule for 3

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

Example of divisibility rule for 3:

Consider the number 43533. The sum of the digits is 4+3+5+3+3, i.e., 18. As 18 is divisible by 3, the number 43533 is divisible by 3.

Divisibility Rule for 9

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

Example of divisibility rule for 9:

Consider the number 43533. The sum of the digits is 4+3+5+3+3, i.e., 18. As 18 is divisible by 9, the number 43533 is divisible by 9.

Divisibility Rule for 6

A number is divisible 6 if it is divisible by both 2 and 3.

Example of divisibility rule for 6:

Consider the number 47394. Since its units digit is 4, it is divisible by 2. Also, its sum of digits is 4+7+3+9+4=27. As 27 is divisible by 3, 47394 is divisible by 3. Hence, 47394 is divisible by 6.

Divisibility Rule for 4

A natural number is divisible by 4 if the number formed by its last two digits in the same order (ten’s digit and unit’s digit) is divisible by 4.

Example of divisibility rule for 4:

34932, 84916, 500 are all divisible by 4 as the numbers formed by taking the last two digits in each case is divisible by 4.

Divisibility Rule for 8

A number is divisible by 8 if the number formed by its last three digits in the same order (hundreds, tens and units digits) is divisible by 8.

Example of divisibility rule for 8:

48640, 47840 and 83400 are all divisible by 8 as the numbers formed by the last three digits in each case are divisible by 8.

Divisibility Rule for 11

A number is divisible by 11 if the difference between the sum of its digits in odd places and the sum of the digits in even places is either 0 or a multiple of 11. 

Example of divisibility rule for 11:

Consider the number 54322455.  

Now, (the sum of digits in odd places) – (the sum of digits in even places) 

=(5+3+2+5)−(4+2+4+5)=0,

which is divisible by 11. 

Hence, 54322455 is divisible by 11. 

1) Test the divisibility of 360 by 2, 3, 4, 5, 6, 8, 9, 10 and 11. 

Divisibility Chart

2	In 360, the unit’s digit is 0, so the number is even. Hence, 360 is divisible by 2.
3	The sum of the digits 3+6+0=9. As 9 is divisible by 3. So, 360 is divisible by 3.
4	The number formed by the last two digits is 60. As 60 is divisible by 4. So, 360 is divisible by 4.
5	In 360, the unit’s digit is 0. So, 360 is divisible by 5.
6	Since 360 is divisible by 2 and 3, it is divisible by 6.
8	The number formed by the last three digits is 360. Since 360/8=45, 360 is divisible by 8.
9	The sum of the digits is 3+6+0=9, so 360 is divisible by 9.
10	In 360, the unit’s digit is 0. So, 360 is divisible by 10.
11	In 360, the sum of the digits in even places is 3+0=3, and the sum of the digits in odd places is 6=6. If you subtract 6 and 3, you get 3. Since 3 is not divisible by 11, 360 is not divisible by 11.

Properties of Divisibility

There are some divisibility properties that help us determine the divisibility of a number by some other numbers.
Property 1: When a number is divisible by another number, it is also divisible by each of the factors of the number.
Example: 36 is divisible by 12 and 2, 3 and 4 are the factors of 12.
⇒ 36 is also divisible by 2, 3 and 4. 
i.e., 36÷2=18, 36÷3=12 and 36÷4=9.   

Property 2: When a number is divisible by two or more co-prime numbers, it is also divisible by its products.
Example: 70 is divisible by 2 and 5 as 70÷2=35 and 70÷5=14.
Since 2 and 5 are co-prime numbers, therefore 70 must be divisible by the product of 2 and 5, i.e.,2×5=10.
⇒ 70 is also divisible by 10. 
i.e., 70÷10=7.

Property 3: When a number is a factor of each of the two given numbers, it is also a factor of their sum and difference.
Example: 5 is a factor of 75 since 75÷5=15.
5 is a factor of 30 since 30÷5=6.
Also, 5 is a factor of (75+30)=105, since 105÷5=21and, 5 is a factor of (75−30)=45, since 45÷5=9.

Property 4: When a number is a factor of another number, it is also a factor of any multiple of that number.
Example: 3 is a factor of 15, since 15÷3=5.
Also, 90 is a multiple of 15 as 15×6=90.
⇒ 3 is also a factor of 90 since 90=3×30.

---

Divisibility Rules – Examples

Example 1

Are the following numbers divisible by 10?
i. 8934
ii. 46120

i. The unit’s digit in 8934 is 4, so by the divisibility test for 10, the number 8934 is not divisible by 10. 
ii. The unit’s digit in 46120 is 0, so by the divisibility test for 10, the number 46120 is divisible by 10. We can verify this by dividing 46120 by 10. 

Example 2

Is 3678 divisible by 2?
First, we check if the unit’s digit is even.

Since the unit’s digit is the even number 8, 3678 is divisible by 2.  

Example 3

Is 129857 divisible by 3?
First, we check if the sum of all the digits is divisible by 3. 

1+2+9+8+5+7=32→no

Since 32 is not divisible by 3, so using the divisibility test for 3, we conclude that 129857 is not divisible by 3.

Example 4

Test if 918 is divisible by 9. Why or why not?
To figure this out, use the divisibility test for 9. Check to see if the sum of the digits is divisible by 9. 

9+1+8=18

18 is divisible by 9; therefore, 918 is also divisible by 9.

Example 5

Is 6162 divisible by 6?
The last digit in 6162 is 2, so by the divisibility test for 2, 6162 is divisible by 2. 
We have 6+1+6+2=15, and 15=3×5, so by the divisibility test for 3, 6162 is divisible by 3. 
Hence, since 6162 is divisible by both 2 & 3, by the divisibility test for 6, 6162 is divisible by 6.

Example 6

Is the number 9812624 divisible by 4? Is it divisible by 8?

The last two digits of 9812624 are 24. \(\frac{24}{4}=6\), so 24 is a multiple of 6.

By the divisibility test for 4, 9812624 is divisible by 4.

The last three digits of 9812624 are 624. \(\frac{624}{8}=78\), so 624 is a multiple of 8. 

By the divisibility test for 8, 9812624 is divisible by 8. 

Example 7

Using the divisibility test for 11 show that 195426 is divisible by 11. 

Sum of the digits in the even place=9+4+6=19

Sum of the digits in the odd place=1+5+2=8

Difference between the two sums=19−8=11

Hence, by the divisibility test for 11, 195426 is divisible by 11. 

Example 8

Find three factors of 31098 using different divisibility tests.

The last digit in 31098 is 8, so 31098 is divisible by 2. 

The last two digits in 31098 are 98, which is not a multiple of 4, so the number is not divisible by 4. Hence the number will also not be divisible by 8. 

Adding the digits in the number, we get 3+1+0+9+8=21,  and \(\frac{21}{3}=7\), so by the divisibility test for 3, we have 31098 is divisible by 3.

Since 31098 is divisible by 2 and 3, by the divisibility test for 6, it is also divisible by 6. 

So, 2, 3 and 6 are three factors of 31098. 

 Remember this! 

• If the unit’s digit of a number is 0, then the number is divisible by 10. 
• If the unit’s digit of a number is 0 or 5, then the number is divisible by 5. 
• If the unit’s digit of a number is 0, 2, 4, 6 or 8, then the number is divisible by 2. 
• A number is divisible by 3 if the sum of its digits is divisible by 3. 
• A number is divisible by 9 if the sum of its digits is divisible by 9. 
• A number is divisible by 6 if it is divisible by both 2 and 3. 
• A number is divisible by 4 if its last two digits are divisible by 4. 
• A number is divisible by 8 if its last three digits are divisible by 8. 
• A number is divisible by 11 if the difference between the sum of alternate pairs of digits is divisible by 11.