**Divisibility Rules**

The divisibility rules or divisibility tests are the math tricks which have made the calculations easier and time saving. Most of the students afraid of doing divisions of huge number digits, so when we learn all these divisibility rules we will be perfect in doing divisions.

Mathematics is the somewhat fun but only we have to learn it passionately. It’s simple to divide two digit number or three digit number by any number but it is too lengthy or tricky to divide more than 4 or 5 digit number by any number that’s why we have to learn divisibility rules.

There are mainly basic 13 rules for test of divisibility starting from 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13. These are the basic divisibility rules which must have to learn by every student. All these divisibility rules are explained with easy examples below. These rules help to us in calculations, numerical problems, and arithmetic problems and so on.

And by applying these rules we get remainder as zero and the perfect quotient. We can apply these divisibility rules only to those numbers which are completely divisible by remaining remainder as zero.

**Divisibility rule for 1:**

This rule states that every number is divisible by one and by dividing with one we get the same number as quotient. That means no need to check divisibility test for one. The number with any digits can also be divisible by one.

__For example__:

- If we take two digit number as 15, if we divide 15 by 1 then we get the quotient as 15 and remainder as zero. That means, when we divide any number by 1 we get the same number as quotient.

- Again, if we divide 300 by 1 then also we get the quotient as 300 and remainder as zero.

Hence, no need to check divisibility of 1 for any number as each number is divisible by one.

**Divisibility rule for 2:**

This rule states that, all the even numbers are divisible by 2. But for 2 or 3 digit number we can identify as even or odd number. When 4 or more than 4 digit number is given then it is not easy. That time we have to check the last one digit i.e.unit digit of that number only. If the last unit digit number is divisible by 2 then that whole number is also divisible by 2.

__For example:__

- If we take a number as 20, we can easily say that it is an even number and hence divisible by 2. Also if we take 80 we fastly say that it is also divisible by 2 as it is even number. This trick is easy up to 2 digit number only.
- But when we take the number as 23456, which is a five digit number and we have to check the divisibility of 2 for this number. It is not easy to decide these number is even or odd. Sometimes it takes time also to identify. Hence here we have to use divisibility rule for 2.
- Now, in 23456 the last unit digit number is 6 that means 6 is divisible 2 so whole number 23456 is also divisible by 2. Thus we have made it easy.
- Similarly for number 890238, here 8 is the last unit digit which is divisible by 2. Hence, the number 890238 is also divisible by 2.
- But if we take the number, 234567, here 7 is the last digit which is not divisible by 2 and hence the whole number 234567 is also not divisible by 2.

**Divisibility rule for 3:**

This rule state that, when the sum of all digits in a number is divisible by 3 then the whole number will be divisible by 3. So we have to add all the digits in number to check the divisibility for 3. Directly we cannot decide the test of divisibility. And this rule is applicable for any number whose quotient will be a perfect or whole number.

__For example:__

- If we take two digit number as 93, the digits in 93 are 9 and 3 only.

So to check divisibility of 3 we add the digits 9 and 3 i.e. 9+3 = 12.

And 12 is divisible by 3 so we can say that 93 is divisible by 3 also.

- If we take the number as 123, the digits in number 123 are 1, 2, and 3.

To check divisibility of 3 for the number 123, we have to find the sum of all digits in it.

Hence, 1 + 2+ 3 =6, and the sum 6 is divisible by 3. Thus, the number 123 is also divisible by 3.

- If we take the number as 145, the total digits in number 145 are 1, 4 and 5.

To check the divisibility of 3 for the number 145, we have to find the sum of all digits.

Hence, 1 + 4 + 5 = 10, and 10 is not divisible by 3.

Thus, the number 145 is also not divisible by 3.

**Divisibility rule for 4:**

This rule states that, if the two last digits in a given number are divisible by 4 then that whole number is also divisible by 4. So for test of divisibility of 4 we have to check last two digits only. And we have to note that, every number divisible by 2 is need not to divisible by 4 also.

**NOTE: **But all numbers which are divisible by 4 are divisible by 2 also.

Such as 8 is divisible by 4 and hence also divisible by 2.

But, 10 is divisible by 2 but not divisible by 4.

__For example:__

- If we have taken the number as 236 is a three digit number, and the last digits are 36. And 36 is divisible by 4, so the whole number 236 is divisible by 4.
- If we have taken the number like 2314, in which last two digits are 14 and it is not divisible by 4. Hence, the whole number 2314 is also not divisible by 4.
- But, in 2314 last two digits 14 is divisible by 2.
- If we have taken the number 1056, in which the last two digits are 56 and 56 is divisible by 4. So the total number 1056 is divisible by 4 also. We don’t have to check the total number, only we have to check last two digits.

**Divisibility rule for 5:**

The divisibility test for 5 is most easy. If we saw the table of 5, then it is observed that the last digit occurring are the only 0 and 5. So we conclude here that, according to test of divisibility for 5, if the last digit of the number is 0 or 5 then it is confirm that the whole number is also divisible by 5.

This is the most easy divisibility test and we can apply it for any digit number also.

__For example:__

- The numbers 25, 35, 20, 255, 800, 670 all are having last digit as either 0 or 5. Hence all the numbers are divisible by 5. No need to check other things here.
- If we have taken the numbers such as 43, 67, 89, 56 etc. these are not having last digit as 0 or 5. And hence these numbers are not divisible by 5.

**Divisibility rule for 6:**

To check the divisibility of 6, we have to first check the divisibility rules for 2 and 3 both. If these divisibility for 2 and 3 both are satisfied then only we say that the number is divisible by 6.

Reminding the rules, if last digit is divisible by 2 then that complete number is also divisible by 2. And if the sum of all digits in a number is divisible by 3 then that complete number is also divisible by 3.

And if above both rules are satisfied by a number then that complete number is divisible by 6.

**NOTE:** For divisibility of 6, first we have to check divisibility for 2 and 3 also. If both are satisfied then only divisibility of 6 is satisfied. Otherwise, if any one of the divisibility i.e. 2 or 3 is not satisfied then it is not divisible by 6.

__For example:__

- If we have taken the number as 516.

For test of divisibility of 2: the last digit in 516 is 6 and 6 is divisible by 2. Hence 516 is also divisible by 2.

For test of divisibility of 3: in the number 516 the digits are 5, 1, 3 and their sum is 5+1+6= 12, and 12 is divisible by 3. Hence the number 516 is divisible by 3.

Thus, the number 516 satisfies both the divisibility tests of 2 and 3. Hence 516 is divisible by 6.

- If we have taken the number 315, in this last digit is 5 which is not divisible by 2 hence no need to check the divisibility for 3. Thus the number 315 is not divisible by 6.

- If we have taken the number 154, in 154 the last digit is 4 which is divisible by 2. Hence the number 154 is also divisible by 2.

Now, in number 154, the digits are 1. 5, 4 and their sum is 1+ 5+ 4 = 10, but 10 is not divisible by 3.

Hence, here test of divisibility for 3 fails. Hence the number 154 is not divisible by 6.

**Divisibility rule for 7: **

To check the divisibility test for 7, we have to subtract the 2 times last digit i.e. unit digit from the rest of the numbers, and then the answer we got if divisible by 7 then that whole number is also divisible by 7. This can be easily explained by using the following examples.

__For example:__

- If we have taken the number 182, in this number the unit digit is 2 and remaining number is 18. So we write for divisibility of 7as,
- 18 – 2 times unit digit
- 18-2*2 = 18-4=14

The answer we got is 14 and which is divisible by 7. Hence, the number 182 is also divisible by 7 here.

- If we have taken the number as 483, in this number the last digit is 3 and remaining number is 48.

So we write here as for divisibility of 7,

48-2*unit digit = 48– 6 = 42

So the answer here we got is 42 and 42 is divisible by 7. Hence the total number 483 is also divisible by 7.

- If we have taken the number as 724, here the unit digit is 4 and the remaining number is 72. To check the divisibility of 7 we write as,

72-2*unit digit number = 72- 2*4 =72- 8 = 64

Here the answer we got is 64 and 64 is not divisible by 7. Hence the number 724 is not divisible by 7.

**Divisibility rule for 8:**

To check the divisibility of 8 of any number we have to check the last three digits of that number only. If the last three digit number is divisible by 8 then that whole number is also divisible by 8. And this trick is applicable to any digit number. To check it in easy way see the following examples.

__For example:__

- If we have taken the number as 3056, in which the last three digit number is 056.

And 056 is divisible by 8. Hence the total number 3056 is also divisible by 8.

In this way we can use this test for any digit number also.

- If we have taken the number 5678990016, this is a 10 digit number and we have to check the test of divisibility of 8 for this number. Here the last three digit number is 016, and 016 is divisible by 8. Hence the total number 5678990016 is also divisible by 8.

Here you can check on calculator also that the above number is divisible by 8 or not directly.

- If we have taken the number as 5555566677788089, in this number the last three digit number is 089 and it is not divisible by 8. Hence the number 5555566677788089 is not divisible by 8.

In this way, we can check the divisibility of 8 for any digit number by considering last three digit number only.

**Divisibility for 9:**

To check the divisibility for 9, we have to take the sum of all digits in a given number and if that sum is divisible by 9 then the total number is divisible by 9. The test of divisibility for 3 and 9 is same. For 3, the sum of all digits must be divisible of 3 and for 9 it must be divisible of 9.

This test of divisibility can be applied to any digit number and easily we can check the divisibility for 9, see following examples.

__For example:__

- If we have taken the number as 585, here the digits are 5, 8, 5. To check the divisibility of 9 for the number 585 we take the sum of all digits in it and write it as,

5+8+5= 18, and the sum 18 is divisible by 9. Hence the number 585 is also divisible by 9.

- If we have taken the number as 1234566, the digits in it are 1, 2, 3, 4, 5, 6 and 6. To check the divisibility of 9 for this number we take the sum of all digits in it and we write it as 1+2+3+4+5+6+6= 27. And the sum we got is 27 which is divisible by 9. Hence, the number 1234566 is also divisible by 9. So this test for 9 is also easy for any digit numbers.

If we have taken the number as 1234567, the digits in it are 1, 2, 3, 4, 5, 6 and 7. To check the divisibility of 9 for this number we take the sum of all digits in it and we write it as 1+2+3+4+5+6+7= 28. And the sum we got is 28 which is not divisible by 9. Hence, the number 1234567 is also not divisible by 9.

**Divisibility rule for 10:**

The easiest test of divisibility is the test of divisibility for 10. Because we have to check the last unit digit only, if the last unit digit is 0 then that any digit number is divisible by 10. So no need to check the other digits here.

__For example:__

- If we have taken the number as 180, here the last digit is 0. Hence the number 180 is divisible by 10. So no need to check the other digits. We have to check only the last digit which must be 0.

- If we have taken the number as 334455667788990, in this number the last unit digit is 0. Hence the complete number 334455667788990 is also divisible by 10.

In this way we can check the divisibility of 10 for any digit number only by seeing the last unit digit which must be 0.

If we have taken the number as 3456789, here the last unit digit is 9 and it is not the 0. Hence the number 3456789 is not divisible by 10.

**Divisibility rule for 11:**

The test of divisibility for 11 is somewhat lengthy but easy. In this test, we have to take difference between the sum of digits which are at even places and the sum of digits which are at the odd places. If this difference is divisible by 11 then that complete number is also divisible by 11. This test is also applied for any digit number.

**NOTE****:** To check even or odd places of the digits we have to give numbering from left to right side only.

__For example:__

- If we have taken the number as 1727, in this number the digits 1 and 2 are at odd places i.e. at 1
^{st}and 3^{rd}positions respectively. But the digits 7 and 7 are at even places ie. At 2^{nd}and 4^{th}positions respectively.

Hence, for test of divisibility of 11 we take sum of digits at even place and sum of digits at odd places.

Sum of digits at even places = 7+7= 14

Sum of digits at odd places = 1+2=3

Hence, difference= sum of digits at even places – sum of digits at odd places

Difference= 14-3= 11

Here the difference we got is 11, and it is divisible by 11. Hence the number 1727 is also divisible by 11.

- If we have taken the number as 185688, here the digits 1, 5, 8 are at odd places. And the digits 8, 6 and 8 are at even places.

Hence for the test of divisibility for 11 we write as,

Difference= sum of digits at even places – sum of digits at odd places

Difference = (8+6+8) – (1+5+8)

Difference= 22- 14= 8

Here we got the difference as 8 and 8 is not divisible by 11. Hence the number 185688 is also not divisible by 11.

**Divisibility rule for 12:**

To check the divisibility rule of 12 for any number then that number must have to satisfy the divisibility test for 3 and 4 also. If both 3 and 4 test are satisfied by that number then it is also divisible by 12. But if any of the divisibility test of 3 or 4 is not satisfied by that number then the divisibility test for 12 fails for that number.

For test of divisibility of 3, we have to take the sum of all digits in that number, if that sum is divisible by 3 then the total number is also divisible by 3.

And, for the test of divisibility of 4 we have to check the last two digits only, if the last two digit number is divisible by 4 then the total number is also divisible by 4.

In this way, if both divisibility of 3 and 4 are satisfied then that total number is also divisible by 12.

__For example:__

- If we have taken the number as 23436, here the digits are 2, 3, 4, 3, 6.

Hence, for test of divisibility of 3, we take the sum of all digits as

Sum = 2+3+4+3+6= 18

Here we got the sum as 18 which is divisible by 3. Hence the total number 23436 is also divisible by 3.

Now, for the test of divisibility of 4, we take last two digit number which is here 36, and 36 is divisible by 4. Hence the total number 23436 is also divisible by 4.

Thus, both the divisibility of 3 and 4 are satisfied by the number 23436.

Hence, 23436 is also divisible by 12.

- If we have taken the number as 13412, here the digits are 1, 3, 4, 1, 2.

To check the divisibility of 3 we take the sum of all digits as,

Sum = 1+3+4+1+2= 11

Here we got the sum as 11, and 11 is not divisible by 3. Hence the total number 13412 is not divisible by 3.

Now, as test of divisibility for 3 fails then no need to check the divisibility for 4.

Hence, the number 13412 is not divisible by 12.

**Divisibility rule for 13:**

To check the divisibility of 13, we have to add 4 times the last digit number i.e. unit digit number from a given number into the remaining number. And this process we have to continue till we get the sum as two digit number. Finally when we get the two digit number then if that two digit number is divisible by 13 then the total number is also divisible by 13.

To see and understand in easy way se the following example.

__For example:__

If we have taken the number as 637, in which the last unit digit is 7 and the remaining number is 63. So to check the divisibility of 13 we write as,

Sum = remaining number+4 times last unit digit number

Sum = 63+ 4*7= 63+28=91

And the sum we got is 91 which is divisible by 13. And hence the total number 637 is also divisible by 13.

In this example we get the sum directly as two digit number, if that sum is not two digit number then we have to repeat the same procedure till we get the sum in two digit form.

If this two digit sum is divisible by 13 then total number is also divisible by 13 otherwise the divisibility test for 13 fails.