Divisibility Rules or Divisibility Tests

 

What is "Divisibility By" ?

 

Divisible By means "When you divide one number by another number, the resultant number is a whole number"

 In other words, if a number is completely divisible by another number, then the quotient will be a whole number and the remainder will be zero.

Examples

• 10 is divisible by 5, because 10 ÷ 5= 2 where as the quotient 2 is a whole number 
• 25 is not divisible by 2, because 25 ÷ 2 = 12.5 where as the quotient 12.5 is not a whole number

The Divisibility Rules

Divisibility rules helps us to test whether one number is divisible by another or not, without the traditional method of division! Sounds Great?

The Divisibility rules from 2 to 19 are given below.

1) Divisibility by 2

If a number is even or a number whose unit digit is either 0, 2,4,6 or 8, it is completely divisible by 2.

Examples:

  1)  7778 is an even number (Here, Unit digit-8)and is completely divisible by 2 

  2) 5665 is not an even number(Here, Unit digit- 5) and hence it is not divisible by 2. 

Steps to check whether 7778 is divisible by 2 or not is as follows:

• Consider the number 7778
• Check whether it is an even number (Check whether the unit digit is either 0 or 2 or 4 or 6 or 8)
• If unit digit is even, then the given number is divisible by 2. 
• Here,8 is the unit digit (Even). Hence, 7778 is divisible by 2.

2) Divisibility Rules for 3

If the sum of its digits is divisible by 3, then the given number is completely divisible by 3.

Foe Example,Consider a number,8361. 

Find the sum of the digits (i.e. 8+3+6+1= 18). 

Now check whether the sum is divisible by 3 or not. 

If the sum is a multiple of 3, then the original number is also divisible by 3. Here, As 18 is divisible by 3, 8361 is also divisible by 3.

Similarly, 4724 is not divisible by 3 completely as the sum of its digits is 17 (i.e. 4+7+2+4=17),which is not a multiple of 3, 17 is not a multiple of 3, therefore 4724 is not a multiple of 3.

3) Divisibility by 4

If the last two digits (Tens and Unit place digits)of a number are divisible by 4, then the given number is a multiple of 4 and is completely divisible by 4 .

Example:

Consider the number 63148. 

Consider the last two digits i.e.48. 

As 48 is divisible by 4, the number 63148 is also divisible by 4.

Similarly, 18430 is not divisible by 4 completely as the last two digits 30 is not a multiple of 4 and also, 18430 is not divisible by 4

4) Divisibility by 5

Numbers with unit digit as 0 or 5 is always divisible by 5.

Example: 20125, 18806520, 995 and 396524850 are divisible by 5

2098, 209857, 3433 and 212 are not divisible by 5.

5) Divisibility by 6

Numbers which are divisible by both 2 and 3 are divisible by 6 (as 2 and 3 are factors of 6 which are co-prime to each other)

If the unit digit of the given number is even and the sum of its digits is a multiple of 3, then the given number is also a multiple of 6.

Example: 930, the number is divisible by 2 as the unit digit is 0.
The sum of digits is 9+3+0 = 12, which is divisible by 3.
Hence, 930 is divisible by 6.

6) Divisibility by 7

In order to check divisibility by 7, First Remove the last digit(Unit digit), multiply it by 2 and subtract it from the rest of the number;if the answer is divisible by 7 , then the number is also divisible by 7(We can apply this rule to that answer again until we find wheather the number is divisible by 7 are not)

Is 1073 divisible by 7?

• Remove 3 from the number and multiply it by 2, it  becomes 6.
• Remaining number is 107, 
• Subtract 6 from 107, 107-6 = 101.
• Repeat the process one more time, last digit is 1 and multiplying with 2 --> 1 x 2 = 2.
• Remaining number is 10; 
• Subtract 2 from 10 --> 10 – 2 = 8.
• As 8 is not divisible by 7, hence the number 1073 is not divisible by 7.

One more example,  

Is 1813 divisible by 7?

• Remove 3 from the number and multiply it by 2, it  becomes 6.
• Remaining number is 181, 
• Subtract 6 from 181, 181-6 = 175.
• Repeat the process one more time, last digit is 5 and multiplying with 2 --> 5 x 2 = 10.
• Remaining number is 17; 
• Subtract 10 from 17 --> 17 – 10 = 7.
• As 7 is divisible by 7, hence the number 1813 is divisible by 7.

7) Divisibility by 8

If the last three digits of a number are divisible by 8, then the number is completely divisible by 8.

Example: 

Take number 20080. Consider the last three digits i.e.  080. As 080 is divisible by 8, the original number 20080 is also divisible by 8.

Another Example,Take number 35206. Consider the last three digits i.e.  206. As 206 is not divisible by 8, the original number 35206 is also not divisible by 8.

8) Divisibility by 9

The rule for divisibility by 9 is similar to divisibility rule for 3.

If the sum of digits of the number is divisible by 9, then the number is also divisible by 9.

Example: Consider 8831, as the sum of its digits (8+8+3+1) is 20, which is not divisible by 9, hence 8831 is not divisible by 9

9) Divisibility by 10

Any number whose unit digit is 0, is divisible by 10.

Example: 100210,51140,65210,792420

10) Divisibility by 11

If the difference of the sum of alternative digits of a number is divisible by 11 i.e. i.e. difference between the digits which are in odd places together and digits in even places together., then that number is divisible by 11 completely.

Check whether the number 57915, is divisible by 11 or not 

• Group the alternative digits i.e. digits which are in odd places together and digits in even places together.
• Take the sum of the digits of each group i.e. 5+9+5=19 and 7+1= 8
• Now find the difference of the sums; 19-8=11
• If the difference is divisible by 11, then the original number is also divisible by 11. 
• Here 11 is the difference which is divisible by 11.
• Therefore, 57915 is divisible by 11.

11) Divisibility by 12

Numbers which are divisible by both 3 and 4 ,then the given number is divisible by 12 (as 3 and 4 are factors of 6)

That is, if the sum of its digits is a multiple of 3 and the last two digits of the given number is divisible by 4, then the given number is also a multiple of 12 i.e. the given number is divisible by 12.

Example: 4824, the last two digits of the number is divisible by 4 as the last two digits is 24, which is divisible by 4.The sum of digits is 4+8+2+4 = 18, which is divisible by 3.
Hence, 4824 is divisible by 12.

12) Divisibility by 13

Remove the last digit, multiply it by 4 and Add it with the rest of the number;

if the answer is divisible by 13 , then the number is also divisible by 13(We can apply this rule to that answer again)

Is 11674 divisible by 13?

• Remove 4 from the number and multiply it by 4, it  becomes 16.
• Remaining number becomes 1167, 
• Add 16 to 1167, 1167+16 = 1183.
• Repeating the process one more time, last digit is 3 and multiplying with 4, 3 x 4 = 12.
• Remaining number is 118; Add 12 with 118--> 118+12 = 130.
• As 130 is divisible by 13, hence the number 11674 is divisible by 13.

13) Divisibility by 14

Numbers which are divisible by both 2 and 7 then, the given number is divisible by 14 (as 2 and 7 are factors of 14)

That is, if the last digit of the given number is divisible by 4 and if it satisfies the divisibility rule of 7 (Have a look on the divisibility rule of 7) , then the given number is also a multiple of 14 i.e. the given number is divisible by 14.

Is 1073 divisible by 14?

Unit digit is 3 which is not an even number, therefore 1073 is not divisible by 2. As it does not satisfy even one of the divisibility rules (divisibility rules of 2 and 7), 1073 is not divisible by 14.

14) Divisibility by 15

Numbers which are divisible by both 3 and 5 , then the given number is divisible by 15 (as 3 and 5 are factors of 15 which are co primes to each other)

That is, if the sum of its digits is a multiple of 3 and the last digit of the given number is either 0 or 5, then the given number is also a multiple of 15.

Example: 930, the sum of digits is 9+3+0 = 12, which is divisible by 3. The number is divisible by 5 as the last digit is 0.
Hence, 930 is divisible by 15.

15) Divisibility by 16

If the last four digits of a number are divisible by 16, then the number is completely divisible by 8.

Example: Take number 20080. Consider the last four digits i.e.  0080. As 0080 is divisible by 16, the original number 20080 is also divisible by 16.

16) Divisibility by 17

Remove the last digit, multiply it by 5 and subtract it from the rest of the number;

if the answer is divisible by 17 , then the number is also divisible by 17(We can apply this rule to that answer again)

Is 1666 divisible by 17?

• Remove unit digit 6 from the number and multiply it by 5, it  becomes 30.
• Remaining number becomes 166, 
• Subtract 30 from 166, 166-30 = 136.
• Repeating the process one more time, last digit is 6 and multiplying with 5, 6 x 5 = 30.
• Remaining number 13~30 = 17.
• As 17 is divisible by 17, hence the number 1666 is divisible by 17.

17) Divisibility by 18

Numbers which are divisible by both 2 and 9 , then the number is divisible by 18 (as 2 and 9 are factors of 18 which are co prime to each other)

That is, if the last digit of the given number is even and the sum of its digits is a multiple of 9, then the given number is also a multiple of 18.

Example: 6678, the number is divisible by 2 as the last digit is 8.
The sum of digits is 6+6+7+8 = 27, which is divisible by 3.
Hence, 6678 is divisible by 18.

18) Divisibility by 19

Remove the last digit, multiply it by 2 and Add it with the rest of the number;

if the answer is divisible by 19 , then the number is also divisible by 19(We can apply this rule to that answer again)

Is 722 divisible by 19?

• Remove the unit digit 2 from the number and multiply it by 2, it  becomes 4.
• Remaining number becomes 72, 
• Add 4 to 72, 72+4 = 76.
• Repeating the process one more time, last digit is 6 and multiplying with 2, 6 x 2 = 12.
• Remaining number is 7; Add 7 with 12 --> 7+12 = 19.
• As 19 is divisible by 19, hence the number 722 is divisible by 19