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Aptitude Tests

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Divisibility Rules - Concept

# Aptitude Tests

-## Divisibility Rules - Concept

Divisibility
rules are useful to find whether a number is exactly divisible by another
number or not without performing actual division.

**Divisibility Rule For 0:**

*Division by zero is not defined*

**Divisibility Rule For 1:**

*Every number is divisible by 1*

**Divisibility Rule for 2**

__:__*In a number, if the last digit is 0, 2, 4, 6, 8 i.e., divisible by 2, then the number is divisible by 2.*

Example: 123456, 208, 304…

**Divisibility Rule for 3:**

*If the sum of all digits of a number is divisible by 3 then the number is also divisible by 3.*

Example
:- 1731

In the
number 1731, the sum of the digits is 1+7+3+1=12, which is exactly divisible
by 3.So 1731 is also divisible by 3.

Example: 825, 1362, 1233…

**Divisibility Rule for 4**

*:*

*If the last two digits of a number are zeroes or the number formed by the last two digits of a number is divisible by 4, then the number is also divisible by 4.*

Example: 314

**28**, 443**00**….**Divisibility Rule for 5:**

*If the last digit of a number is 0 or 5, then the number is exactly divisible by 5.*

Example: 40

**0**, 40**5**, 105**5**…**Divisibility Rule for 6:**

*If a number is exactly divisible by both 2 and 3, then the number is also divisible by ‘6’.*

Example: 324, 1254

**Take the number 216,**

The given number is even number, so it is divisible by 2

And the sum of the digits is 2+1+6 =9, which is multiple of 3.

So the
number 216 is multiple of both 2 and 3, so the number is exactly divisible
by 6.

**Divisibility Rule for ‘7’:**

Take the last digit of the number, multiply it by 2, and subtract the product from the rest of the number. If the answer is divisible by 7 (including 0), then the number is also divisible by 7. Continue this until you get a one-digit number. The result is 7, 0, or -7, if and only if the original number is a multiple of 7.

*In aptitude tests,*we usually don't find the questions on*divisibility by 7.*But it will be helpful to do some**quick calculations.**Take the last digit of the number, multiply it by 2, and subtract the product from the rest of the number. If the answer is divisible by 7 (including 0), then the number is also divisible by 7. Continue this until you get a one-digit number. The result is 7, 0, or -7, if and only if the original number is a multiple of 7.

Example: 371

371×2

-2

------

35

35 is
divisible by 7. The given number 371 is also divisible by 7

**Divisibility Rule for ‘8’**

**:**

*If the number formed by the last three digits of a number is divisible by ‘8’ or last three digits of the given number are zeroes then the number is divisible by ‘8’.*

*Example: 56847000, 9432*

**256**
Example: In the number 27358128, the number formed by the last three digits is 128.

128 is exactly divisible by 8, then the number 27358128 is exactly divisible by 8

128 is exactly divisible by 8, then the number 27358128 is exactly divisible by 8

**Divisibility Rule for ‘9’:**

*If the sum of all digits is a number is divisible by 9, then the number is divisible by 9.*

Example: 172845.

Sum of the digits => 1+7+2+8+4+5 =27 which is a multiple of 9

So the
number 172845 is also divisible by 9.

**Divisibility Rule for 10:**

In the
given number the last digit is ‘0’ then the number is divisible by 10.

Example: 1000, 2120…

**Divisibility Rule for ‘11’**

*:*

*A number is divisible by 11. If the difference of the sum of its digits at odd places and sum of its digits at even places is either 0 or a number divisible by 11.*

Example 1 :
14641,

(Sum of digits at odd places) – (sum of digits at even places)

(Sum of digits at odd places) – (sum of digits at even places)

(
1 + 6 + 1 ) - ( 4 + 4 ) = 0

The
difference is 0, the number 14641 is exactly divisible by 11

Example 2 :

The
number 4832718 is divisible by 11
, Since

(Sum of
digits at odd places) – (sum of digits at even places)

( 8 + 7 + 3 + 4 ) - ( 1 + 2 +
8 ) = 11 which is divisible by 11.

**Divisibility Rule for ‘12’:**

*If a number is exactly divisible by both 3 and 4, then the number is divisible by 12*

Example:
1752

In this
number, the sum of the digits is 1+7+5+2=15 . 15 is multiple of 3, so the number is multiple of 3

1752,
last 2 digits 52 exactly divisible by 4, so the number is divisible by 4

So the
number 1752 is divisible by both 3 and 4, the number is divisible by 12

**Divisibility Rule for ‘13’:**

*In*

Take the last digit of the number, multiply it by 4, and add the product to the rest of the number. If the answer is divisible by 13, then the number is also

*aptitude tests*, usually, questions on divisibility by 13 do not appear. But here we discussed because of it useful for f*aster calculations.*Take the last digit of the number, multiply it by 4, and add the product to the rest of the number. If the answer is divisible by 13, then the number is also

*divisible by 13*. Continue this until you get a two-digit number. The result is multiple of 13, if and only if the original number is a multiple of 13.
Example: 1391
=> 1391 x 4

__+ 4__

143

As 143 is
multiple of 13, the number 1391 is multiple of 13

**Divisibility Rule for ‘14’:**

*If a number is exactly divisible by both 2 and 7, then the number is exactly divisible by 14*

Example: 1512
, The number is even, so it is divisible by 2

Check
it by 7, 1512*2

__-4__

147

The remainder 147 is exactly divisible by 7.

The given
number 1512 is exactly divisible by both 2 and 7, so the number is
divisible by 7

**Divisibility Rule for ‘15’**

*:*

*If a number multiple of both 3 and 5, the number is multiple of 15*

Example: 5415

In this
number, the sum of the digits is 5+4+1+5=15 . 15 is multiple of 3, so the number is multiple of 3

5415, the last digit is 5, so the number is divisible by 5.

The
number 5415 is divisible by both 3 and 5, so the number is
divisible by 15

**Divisibility Rule for ‘16’**

*:*

*If last 4 digits of a number are zeroes or multiple of 16, then the number is multiple of 16*

Example
1: 72568

**0000**Here last 4 digits are 0's, so the number is divisible by 16.
Example 2:: 7293181680, In this number
last 4 digits are 1680, is a multiple of 16. So the number is multiple of 16.

**Divisibility Rule for 17**

*Multiply the last of digit of the given number by 5, and then subtract the product from remaining truncated number. Repeat the step as necessary. If the result is divisible by 17, the original number is also divisible by 17*

Example
: Take the number 1938.

193 8 x5

-40

----

153.

Since 153
is divisible by 17, the original number 2278 is also divisible.

**Divisibility Rule for 18.**

*If a number is divisible by both 2 and 9, then the number is exactly divisible by 18*

Example
: 23526

23526 is
an even number, so it is divisible by 2

The sum of
the digits of the number 23526 is ( 2+3+5+2+6)=18 which is divisible by 9.

So 23526
is exactly divisible by both 2 and 9, the number is divisible by 18.

**Divisibility Rule for 19.**

*MultiplY the last digit of the original number by 2 and then add the product to the remaining truncated number. Repeat the step as necessary. If the result is divisible by 19, the original number is also exactly divisible by 19*

Example :
Let us check for 3895::

389 5x2

+10

-----

39 9x2

+18

-----

57

Since 57 is divisible by 19, original number 3895
is also divisible by 19.

For
solved problems and examples on Divisibility Rules - go through