Divisibility Rules

 

So, how do we check if a number is divisible by another number (divisor). Do we have to divide the number to check if the reminder is 0 to conclude if the number is divisible by divisor?

If the number or the divisor is a large number, maybe we have to actually divide by taking the factors or dividing using the long division method (paper-pencil technique) .

However, there are few numbers (divisors) for which we do not have to actually divide the number, by just looking at the number or part of the number, we can conclude  whether the number is divisible by the divisor.

In this tutorial, we are going to discuss the Divisibility rules for numbers (divisors) starting from 2.

Divisibility by 1


So, do we need to check if the number is divisible by 1? You know, all the numbers are divisible by 1. So there is nothing to check here, nor do you any shortcut ๐Ÿ™‚

Divisibility by 2


RULE :  Check if the last digit of the number is even (0,2,4,6,8).

Example:

  • 128 is divisible by 2, as 8 is divisible by 2
  • 129 is not divisible by 2, since 9/2 = 4 1/2

 

Exercise : Can you if the following numbers are divisible by 2?

  • 1345
  • 2356
  • 34563
  • 85785

Divisibility by 3

RULE : The sum of the digits is divisible by 3.

As you know, the sum of the digits of a number is nothing but the digital sum of the number. So, if the digital sum of a number is divisible by 3, then the number itself is divisible by 3.

Example:

  • 38145949

Divisibilty3

The digital root of 38145949 is 3+4 =7. 7 is not divisible by 3. So, 38145949 is also not divisible by 3.

  • 7463652

Divisibilty2

The digital root of 7463652 is 6. 6 is divisible by 3. So, 7463652  is also divisible by 3.

 

Exercise : Can you if the following numbers are divisible by 3 ?

  • 134566
  • 23566
  • 34563
  • 895785

Divisibility by 4


RULE :  The last two digits of the number should be divisible by 4.

If you have to check if the number is divisible by 4, ignore all the digits in the number, and just consider the last 2 digits. If this 2-digit number is divisible by 4, then the whole number is divisible by 4.

EXAMPLE:

  • 38145949

By looking at the number itself, we can conclude, that the number is not divisible by 4 as the number is not even. So, we do not have any further. We can conclude 49 is not divisible by 4, also 38145949 is also not divisible by 4.

  • 895788

So, the number is even, so we will check if 88 is divisible by 4. 88 is divisible by 4. So, 895788 is divisible by 4.

Exercise : Can you if the following numbers are divisible by 4 ?

  • 134566
  • 235660
  • 345632
  • 895785

Divisibility by 5


RULE : The last digit of the number should be either 5 or 0.

EXAMPLE:

  • 38145940 is divisible by 5 as the number ends with 0.
  • 55551 is not divisible by 5 as the number ends with 1.

Exercise : Can you if the following numbers are divisible by 5 ?

  • 134566
  • 235660
  • 345632
  • 895785

Very easy right ๐Ÿ™‚

Divisibility by 6


RULE : The number should be divisible by both its prime factors 2 and 3.

If you have to check if the number is divisible by 6. First we will check if the number is divisible by 2 which is very easy to find out. Just check the last digit, if it is even or not.

If the number is even, only then we can check if the number is divisible by 3, then we can conclude that number is divisible by 6.

EXAMPLE:

  • 38145940

By looking at the number itself, we can see that the number is even so it divisible by 2. Now, letโ€™s if the number is divisible by 3. 

Divisibilty3

The digital root of 38145940 is 3+4 = 7. So, 38145940 is not divisible by 6.

  • 834

834 is an even number. Hence it is divisible by 2.  The digital root of 834 is 6 which is divisible by 6. So 834 is divisible by 6.

Exercise : Can you if the following numbers are divisible by 6 ?

  • 134566
  • 235660
  • 345632
  • 895785

Divisibility by 7


There are a number of shortcuts discussed in the literature. This is by far the easiest I found.

  • If a number 21 is divisible by 7, 210 is also divisible by 7, and so is 2100, 21000 etc.
  • If 35 is divisible by 7, then adding/subtracting any multiple of 7 from 35 will also be divisible by 7. So, 35 โ€“ 7 =28 is divisible, 35 โ€“ 14 = 21 is divisible by 7.

Let me explain the shortcut with an example. Letโ€™s say we want to check the divisibility of 3395

 

Divisibilty7_1

 

Step 1 : Subtract or add a multiple of 7 to the number, so that we have 0 at the end. Subtract 35 from 3395. We get 3360.

Step 2 : Discard the 0 at the end. Now we need to check the divisibility of 336.

Step 3 : Add 14 to 336 to get 350 and again discard 0. We get 35.

Now, 35 is divisible by 7. Hence 3395 is also divisible by 7.

Consider another example. Letโ€™s say we want to check the divisibility of 18543

Divisibilty7_3

 

Step 1 : Add 7 to 18453. We get 18550.

Step 2 : Discard 0 from 18550.Now we need to check the divisibility of 1855.

Step 3 : Subtract 35 from 1855 to get 1820. Again discard 0 from 1820.

Step 4 : Subtract 42 from 182 to get 140. Again discard 0 from 140 to get 14.

Now, 14 is divisible by 7. Hence 18543 is also divisible by 7.  For larger numbers, the steps keep on increasing. Sometimes, it is preferable to go for long division method.

Similarly, can you check the divisibility of  4432 ?

 

Divisibilty71

Exercise : Can you if the following numbers are divisible by 7 ?

  • 32662
  • 45168
  • 65758
  • 48678

 

Important : This method can also be used to check the divisibility of  13.

Divisibility by 8


RULE :  The last 3 digits of the number should be divisible by 8.

Consider just the last 3 -digits of the number ignoring the rest of the digits. If this 3-digit number is divisible by 8, then the whole number is divisible by 4.

EXAMPLE:

  • 38145872

The last 3-digits 872 (800+72) is clearly divisible by 8. We can conclude, that the number 38145872 is divisible by 8.

  • 895788

The number 788 is not divisible by 8. So, 895788 is not by 8.

 

If a number is divisible by 8, then that number is also divisible by 2 and 4 as 2 and 4 are the factors of 8. However the reverse is not always true. If a number is divisible by both 2 and 4, then it does not guarantee that the number is divisible by 8 as well. 

In general, a number which is divisible by a divisor is also divisible by the prime-factors of  the divisor.

Exercise : Can you if the following numbers are divisible by 8 ?

  • 134566
  • 235660
  • 345632
  • 895785

Divisibility by 9


RULE :  The sum of the digits is divisible by 9.

As you know, the sum of the digits of a number is nothing but the digital sum of the number. So, if the digital sum of a number is divisible by 9, then the number itself is divisible by 3.

If a number is divisible by 9, then the digital sum of this number is 0.

EXAMPLE:

  • 7463652

Divisibilty2

The digital sum of the number is 6 which is not divisible by 9. So 7463652 is not divisible by 9.

  • 38145949

Divisibilty1

  • The digital sum of the number is 3+4 = 7 which is not divisible by 9. So 7463652 is not divisible by 9.

 

Note : When you are obtaining the digital sum of a number which is perfectly divisible by 9, all the number gets cancelled out.

  • 1629324

Divisibilty4

Notice this number, when you obtaining the digital sum of  1629324, all the numbers are cancelled. So, the digital sum of the number is 0. Hence 1629324 is divisible by 9.

Exercise : Can you if the following numbers are divisible by 9 ?

  • 9233320
  • 29822
  • 5478478
  • 389292

Divisibility by 10


RULE :  Check the last digit of the number. A number which is divisible by 10 will have a 0 as the unit digit

EXAMPLE:

  • 38145940 is divisible by 10
  • 5555 is not divisible by 10  as the number ends with 5.

Note A number which is divisible by 10 is also divisible by 2 and 5 (2 & 5 are the prime factors of 10), vice-versa is not necessarily true.

 

Divisibility by 100


A number which is divisible by 100 will always have last 2 digits as 0. Similarly, A number which is divisible by 1000 will always have last 3 digits as 0.

Divisibility by 11


RULE :

Step 1 :  Starting from 1st digit, add all the alternative digits.

Step 2 :  Add the remaining digits.

Step 3 :  Take the difference of the numbers obtained in Step1 and Step 2. If the difference is 0 or a multiple of  11, then the number is divisible by 11.

EXAMPLE :

  • 5016

Step 1 :  5 + 1 =6

Step 2 :  0 + 6 = 6

Step 3 :  So, 6 โ€“ 6 = 0. Hence 5016  is also divisible by 11.

  • 586564

Divisibilty6

Step 1 :  5 + 6 + 6 = 17

Step 2 :  8 + 5 + 4 = 17

Step 3 :  So 17 โ€“ 17 = 0. Hence 586564  is also divisible by 11.

 

Exercise : Can you if the following numbers are divisible by 11 ?

  • 61039
  • 249705
  • 199764
  • 671429

Divisibility by 12


RULE : A number should be divisible by both 3 and 4

The strategy is to check if the number is divisible by 4 first. By inspection itself, you can check if the number is divisible by 4.

EXAMPLE :

  • 5472

72 is divisible by 4. The digital sum of 5472 is 0. Hence 5472 is divisible by 3 ( also divisible by 9 ). Therefore, we can conclude, 5472 is divisible by 12.

  • 722488

88 is divisible by 4. The digital sum of the number is 2 + 4 + 8 + 8  = 22 = 4. Hence 722488 is not divisible by 3 Therefore, we can conclude, 722488 is not divisible by 12.

Exercise : Can you if the following numbers are divisible by 12 ?

  • 2488860
  • 294552
  • 63644
  • 67752

 

Important : We mentioned earlier, if a number is divisible by a divisor, then the number is also divisible by all its factors.

The factors of 12 are 1,2,3,4,6,12

So, any number which is divisible by 12, is also divisible by 2, 3,4 and 6.

We mentioned in the shortcut, if a number is divisible by 3 and 4, then the number is divisible by 12.  Notice, that the numbers 3 and 4 are co-primes (If 2 numbers have a HCF of 1, then they are called co-primes)

Even though, 2 and 6 are factors of 12, we cannot use the divisibility of 2 and 6 to confirm the divisibility of 12.

For example, 18 is divisible by both 2 and 6. But 18 is not divisible by 12. Because 2 and 6 are not co-primes.

Divisibility by 13

The shortcut to check the divisibility is very similar to divisibility by 7 rule. Except you add/subtract the multiples of 13 here.

 

Divisibility by 16

RULE : Last 4-digits of the number should be divisible by 16

 

Divisibility by 32

RULE : Last 5-digits of the number should be divisible by 32 and so on.

 

Divisibility by 15

RULE : A number should be divisible by both 5 and 3 ( 5 and 3 are co โ€“factors)

 

Divisibility by 18

RULE : A number should be divisible by both 2 and 9

 

Divisibility by  21

RULE : A number should be divisible by both 7 and 3

 

Divisibility by  24

RULE : A number should be divisible by both 8 and 3

Now, 3 and 8 are factors of 24, so is 12 and 2 right.  So, can we conclude if the number is divisible by both 12 and 2, then the number itself is divisible by 24 ?

Answer is No.  Because 12 and 2 are not co-primes.

Consider a very simple example. Consider the number 36. 36 is divisible by both 12 and 2. But it is not divisible by 24.

So, in order to check if the number is divisible by 24, we need to check the divisibility of that number by 3 and 8.

Let me explain with a very simple example. Consider the number 36. 36 is divisible by both 12 and 2. But it is not divisible by 24.

So, in order to check if the number is divisible by 24, we need to check the divisibility of that number by 3 and 8.

Divisibility by  36

Consider the factors of 36:

  • ( 1, 36 )
  • ( 2, 18 )
  • ( 3, 12 )
  • ( 4, 9 )
  • ( 6, 6 )

Notice, that except (4,9) none of the numbers are co-primes. So if the number is divisible by 4 and 9, then it is also divisible by 36.

RULE : A number should be divisible by both 9 and 4.

 

Divisibility by  25

RULE : Last 2-digits of the number should be divisible by 25. So a number which ends with 25, 50, 75, 00 are all divisible by 25

 

Divisibility by  125

RULE : Last 3-digits of the number should be divisible by 125.

 

Divisibility by  50

RULE : Last 2-digits of the number should be divisible by 50. So, the last 2-digits of the number should be either 00 or 50.

 

I hope you find this article on Divisibility Rules useful. I have tried to cover all the rules for most of the numbers.  Please contact me if some point in the tutorial is not clear. I would be glad to help.

Note : If you like the article, please leave a comment in the comment section. I like reading them to my niece  ๐Ÿ™‚ . Please be nice to her who is all of 3.   All the comments which I consider it as rude will be taken off  ๐Ÿ™‚

 

Kiran Chandrashekhar

Hey, Thanks for dropping by. My name is Kiran Chandrashekhar. I am a full-time software freelancer. I love Maths and Mathematical Shortcuts. Numbers fascinate me. I will be posting articles on Mathematical Shortcuts, Software Tips, Programming Tips in this website. I love teaching students preparing for various competitive examinations. Read my complete story.

7 Comments

  1. Hi KIRAN ji… I did read most of your articles on mathematics shortcut methods.. All of them are very helpful for people like us , who prepare for competitive exams… Thank you very much for spending your precious time, experience and intelligence in sharing these useful information for Us.

  2. Hi Sir
    Its rly very useful to me for preparing competitive exam….will u please post all shortcuts for Aptitude book by R.S.Agarwal(if u have)

    • Please ask me some specific question I can help you. R.S.Agarwal is already solved. I think you will find the steps there. If you have doubts please ask me

  3. Sir all ur articles r very interstng & useful for us & i m preparing for comptve. exam. Sir if u hv any shortcuts for reasoning pls share with us, it ll be very useful for us…

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