How do you find whether a number is exactly divisible by
another number? By performing division calculation? No, you cannot do that for
every number. That’s why we need to follow divisibility rules. These are useful to solve problems quickly in Aptitude Tests. These rules are mainly useful to check if a number is Prime. Suppose you want
to find 45647845631214566461 is divisible by 3 or not. You cannot perform
division calculation because it takes a lot of time. This is where you need
divisibility rules. This rules makes your division easy. Here, divisibility rules for different numbers with examples under each divisibility
rule is explained and generalized rules for divisibility by any number is also included at the end.

## Contents :

**1. What is Divisibility**

**2. Divisibility Rules For Numbers Up To 20**

**3. Generalized Divisibility Rules**

### What is Divisibility :

“Divisible by” means when you divide one number with another number the result should be whole number with zero remainder.

Ex :

6/3 = 2 ; 6 is divisible by 3, because result 2 is whole
number and remainder is 0.

7/3 = 2.33 ; 7 is not divisible by 3, because result 2.33 is
not whole number and remainder is 1.

**Co Primes :**H.C.F. of two numbers in one set should be 1.

Ex : (2,3),(4,5),(6,7),(11,15)

If a number N is divisible by two numbers a and b, where a, b are co primes, then N is divisible by ab.

Ex :

24 is divisible by 2 and 3 where 2,3 are co primes,
then 24 is divisible by 6.

36 is divisible by 2, 4 where 2, 4 are not co primes, so, 36
is not divisible by 8.

If a number N is divisible by another number M, then N is also divisible by the factors of M.

Ex:

72 is divisible by 12, 72 is also divisible by 2, 3, 4,
6. because 2, 3, 4, 6 are factors of 12.

If a number N is divisible by prime factors of M, then N is also divisible by M.

Ex:

63 is divisible by 3 and 7, 63 is also divisible by 21,
because 3 and 7 are prime factors of 21.

### Divisibility Rules For Numbers Up To 20 :

#### Divisibility by 1 :

Every integer is divisible by 1, so no rules are needed.

Ex: 3 is divisible by 1, 4573842 is divisible by 1.

#### Divisibility by 2 :

The last digit must be even number i.e. 0, 2, 4, 6, 8.

Ex : 3456 is divisible by 2 –> last digit i.e. 6 is an even number.

343423 is not divisible by 2 –> las

t digit 3 is not an even number.

t digit 3 is not an even number.

#### Divisibility by 3:

The sum of digits in given number should be divisible by 3.

Ex : 3789 is divisible by 3 –> sum 3+7+8+9= 27 is
divisible by 3.

43266737 is not divisible by 3 –> sum 4+3+2+6+6+7+3+7 = 38
is not divisible by 3.

#### Divisibility by 4 :

The number formed by last two digits in given number must be
divisible by 4.

Ex : 23746228 is divisible by 4 –> 28 is divisible by 4.

674235642 is not divisibility by 4 –> 42 is not divisibility by 4.

#### Divisibility by 5 :

Last digit must be 0 or 5.

Ex: 42340 is divisible by 5 –> 0 is last digit.

675564 is not divisibility by 5 –> 4 is last digit.

#### Divisibility by 6:

The number must be divisible by 2 and 3. Because 2 and 3 are
prime factors of 6.

Ex : 7563894 is divisible by 6 –> last digit is 4, so
divisible by 2, and sum 7+5+6+3+8+9+4 = 42 is divisible by 3.

567423 is not divisible by 6 –> last digit is 3, so not
divisible by 2. No need to check for 3.

#### Divisibility by 7:

Twice the last digit and subtract it from remaining number
in given number, result must be divisible by 7. (You can again apply this to
check for divisibility by 7.)

Ex :

1. 343 is divisible 7 –> 34 – (2*3) = 28, 28 is divisible
by 7.

2. 345343 is given number. 3 is last digit. Subtract 2*3
from 34534.

34534 – (2*3) =
34528, we cannot tell that this result is divisible by 7. So, we do it again.

3452 – (2*8) =
3436, we will do it again.

343 – (2*6) = 331,
we will do it again.

33 – (2*1) = 31,
31 is not divisible by 7.

So, 345343 is not
divisible by 7.

Don’t forget to check the generalized rule at the end.

#### Divisible by 8:

The number formed by last three digits in given number must
be divisible by 8.

Ex : 234568 is divisible by 8 –> 568 is divisible by 8.

4568742 is not divisible by 8 –> 742 is not divisible by 8.

#### Divisible by 9:

Same as 3. Sum of digits in given number must be divisible
by 9.

Ex : 456786 is divisible by 9 –> 4+5+6+7+8+6 = 36 is divisible
by 9.

87956 is not divisible by 9 –> 8+7+9+5+6 = 35 is not
divisible by 9.

#### Divisible by 10 :

Last digit must be 0.

Ex : 456780 is divisible by 10 –> last digit is 0.

78521 is not divisible by 10 –> last digit is 1.

#### Divisible by 11 :

Form the alternating sum of digits. The result must be
divisible by 11.

Ex : 416042 is divisible by 11 –> 4-1+6-0+4-2 = 11, 11 is
divisible by 11.

8219543574 is divisible by 11 –> 8-2+1-9+5-4+3-5+7-4 = 0
is divisible by 11.

#### Divisibility by 12 :

Ex : 462157692 is divisible by 12 –> last 2 digits 92, so
divisible by 4, and sum 4+6+2+1+5+7+6+9+2 = 42 is divisible by 3.

625859 is not divisible by 6 –> last 2 digits 59, is not
divisible by 4. No need to check for 3.

#### Divisibility by 13 :

Multiply last digit with 4 and add it to remaining number in
given number, result must be divisible by 13. (You can again apply this to
check for divisibility by 13.)

Ex :

1. 4568 is not divisible by 13 –> 456 + (4*8) = 488 –> 48 + (4*8) = 80, 80 is not divisible by 13.

2. 593773622 is given number. 2 is last digit. add 4*2 to 59377362 -> 59377370

5937737+ (4*0) = 5937737,
we cannot tell that this result is divisible by 13. So, we do it again.

593773 + (4*7) = 593801,
we will do it again.

59380 + (4*1) = 59384,
we will do it again.

5938 + (4*4) = 5954,
we will do it again.

595 + (4*4) = 611, we will do it again.

61 + (4 *1) = 65 , 65 is divisible by 13.

61 + (4 *1) = 65 , 65 is divisible by 13.

So, 593773622 is
divisible by 13.

Don’t forget to check the generalized rule at the end.

#### Divisibility by 14 :

The number must be divisible by 2 and 7. Because 2 and 7 are
prime factors of 14.

####

Divisibility by 15 :

The number should be divisible by 3 and 5. Because 3 and 5
are prime factors of 15.

#### Divisible by 16:

The number formed by last four digits in given number must
be divisible by 16.

Ex : 7852176 is divisible by 16 –> 2176 is divisible by
16.

#### Divisibility by 17 :

Multiply last digit with 5 and subtract it from remaining
number in given number, result must be divisible by 17. (You can again apply
this to check for divisibility by 17.)

Follow the similar examples given in divisibility by 7 and
divisibility by 13.

#### Divisibility by 18 :

The number should be divisible by 2 and 9. Because \(2~\times
~{{3}^{2}}\) are prime factors of 18.

#### Divisibility by 19 :

Multiply last digit with 2 and add it to remaining number in
given number, result must be divisible by 19. (You can again apply this to
check for divisibility by 19.)

Follow the similar examples given in divisibility by 7 and
divisibility by 13.

#### Divisibility by 20 :

The number formed by last two digits in given number must be
divisible by 20.

Ex : 2374680 is divisible by 20 –> 80 is divisible by 20.

456215789654824 is not divisibility by 20 –>
24 is not divisibility by 20.

### Generalized Divisibility rules :

To check divisibility by a number You should check divisibility by highest power of each of its prime factors.

Remember that the divisibility by any two of the factors of
a number is not sufficient to judge its divisibility.

Ex :

1. Suppose you are checking for divisibility by 12. 2,3,4,6
are factors of 12. You cannot say that the number is divisible by 12 by
checking divisibility by only 2 and 4 or 4 and 6. You should check divisibility
by 3 and 4 to tell the divisibility by 12. Because 3 and 4 are prime factors of
12.

2. Now, we are checking divisibility by 18. Prime factors of
18 are \(2~\times ~{{3}^{2}}\). So, you should check
divisibility by 2 and 9 . don’t check for just 2 and 3 and also don’t check for
3 and 6.

Basically prime numbers have factors 1 and the number
itself. So, we can’t this rule apply for prime numbers. This is only for
composite numbers.

####
**Divisibility by numbers which ends in 1,3,7,9 :**

So, this rule counts for prime numbers which have been
missed in previous rule.

To test divisibility by a number N which ends in 1,3,7,9
this method can be used.

Multiply N with any number to get 9 in the end. Add 1 to the result and divide it by 10.

Store the above result as R.

We are checking whether a number X is divisible by N or not.

Split X as X = 10 y + z ;

X is divisible by N, only if Rz + y is divisible by N.

Ex – 1 :

Find whether 645 is divisible by 23 or not.

N =23 ; 23 * 3 = 69 ; so now N has 9 in the end.

R = (69 + 1) / 10 = 7 ;

X = 645 ; split X as X = 10 y + z ;

645 = (10 * 64) + 5 ;

Y = 64 ; z = 5 ;

Rz + y = (7 * 5) + 64 = 35 + 64 = 99 ;

99 is not divisible by 23. So, 645 is also not divisible by
23.

Ex – 2 :

Let us find 585 is divisible by 39 or not.

N =39 ; so now N has
9 in the end.

R = (39 + 1) / 10 = 4 ;

X = 585 ; split X as X = 10 y + z ;

645 = (10 * 58) + 5 ;

Y = 58 ; z = 5 ;

Rz + y = (4 * 5) + 58 = 20 + 58 = 78 ;

78 is divisible by 39. So, 585 is also divisible by 39.

Read Also :

How to find prime numbers and how to check if a number is prime

Basic Number Theory for problems on numbers

Basic Number Theory for problems on numbers