Contents

- 1 Divisibility Rules for easy calculations of mathematics
- 1.0.1 Divisibility rules for 2, 4, 8, 16 :
- 1.0.2 Divisibility rules for 3, 9 :
- 1.0.3 Divisibility rules for 5 :
- 1.0.4 Divisibility rules for 6 :
- 1.0.5 Divisibility rules for 7 :
- 1.0.6 Divisibility rules for 11 :
- 1.0.7 Divisibility rule for 13 :
- 1.0.8 Divisibility rule for 17 :
- 1.0.9 Divisibility rule for 19 :

- 1.1 Number System:

## Divisibility Rules for easy calculations of mathematics

Here given *divisibility rule*s for the numbers from 1 to 20.

#### Divisibility rules for 2, 4, 8, 16 :

**Divisible by 2** : Last digit of a any number is divisible by 2 than that hole number is divisible by 2

i.e Any number end with 0,2,4,6,8 than it is **divisible by 2**

**Divisible by 4** : A number is divisible by 4, if the number formed with its last two digits is divisible by 4.

**Divisible by 8** : A number is divisible by 8, if the number formed with its last three digits is divisible by 8.

**Divisible by 16 :** A number is divisible by 16, if the number formed with its last three digits is divisible by 16.

**Examples:**

1.Check divisibility by 2, 4, 8 & 16 of the number 413984.

- The lost digit of the above number is 4. So this number is divisible by 2.
- Take lost two digits i. e 84 and 84 divisible by 4 ( 84/4 = 21). So this number is divisible by 4
- Now take lost three digits i. e 984 and 984 divisible by 8 ( 984/8 = 123). So this number is divisible by 8.
- Take lost four digits i. e 3984 and 3984 divisible by 16 ( 3984/16 = 249). So this number is divisible by 16.

2. Check divisibility by 2, 4, 8 & 16 of the number 24589436.

- The lost digit of the above number is 6. So this number is divisible by 2.
- Take lost two digits i. e 36 and 36 divisible by 4 ( 36/4 = 9). So this number is divisible by 4
- Now take lost three digits i. e 436 and 436 not divisible by 8 ( 436/8 = reminder 4). So this number is not divisible by 8.
- Take lost four digits i. e 9436 and 9436 is not divisible by 16 ( 9436/16 = reminder 4). So this number is not divisible by 16.

#### Divisibility rules for 3, 9 :

All such numbers the **sum of whose digits are divisible by 3** are divisible by 3. In the same rule also applicable for 9 also i.e the** sum of digits of a number are divisible by 9** then it is divisible by 9.

**Examples :**

1.Check divisibility by 3 & 9 of the number 25498644.

The sum of all digits in a given number = 2+5+4+9+8+6+4+4 = 42

So here 42 is divisible by 3 and not divisible by 9 . So given number is also divisible by 3 and not divisible by 9.

1.Check divisibility by 3 & 9 of the number 232911.

The sum of all digits in a given number = 2+3+2+9+1+1 = 18

So here 18 is divisible by 3 and also divisible by 9 . So given number is also divisible by 3 and 9.

#### Divisibility rules for 5 :

Any number **end with 0 or 5** then it is divisible by 5.

1.Check divisibility of the number 95487430 by 5

The lost digit of the given number is “0” . So it is divisible by 5.

2.Check divisibility by 5 of the number 265474585.

The lost digit of the given number is “5” . So it is divisible by 5.

#### Divisibility rules for 6 :

A number is divisible by 6, if it is **simultaneously divisible by 2 and 3.**

**Example:**

1.Check divisibility by 6 of the number 357282918.

- The lost digit of the above given number is 8. So this number is divisible by 2.
- The sum of all digits in a given number = 3+5+7+2+8+2+9+1+8 = 45 and it is divisible by 3 (45/3=15).
- The given number is divisible by 2 and 3. So it is also divisible by 6.

2.Check divisibility by 6 for the number 4875614.

- The lost digit of the above given number is 4. So this number is divisible by 2.
- The sum of all digits in a given number = 4+8+7+5+6+1+4 = 35 and it is not divisible by 3 (35/3= reminder is 2)
- The given number is divisible by 2 and not divisible by 3. So the given number is not divisible by 6.

#### Divisibility rules for 7 :

We use **oscillator (– 2)** for divisibility test.

for example take number 39732

step 1 : Separate unit place from the given number and multiplied by “-2” for unit digit place ( 2 x -2 = -4)** .** This product is to be add for remaining number [ 3973 + (-4) = 3969].

step 2 : Separate unit place from 3969 and multiplied by “-2” for unit digit place ( 9 x -2 = -18) . This product is to be add for remaining number [ 396 + (-18) = 378].

step 3 : Separate unit place from 378 and multiplied by “-2” for unit digit place ( 8 x -2 = -16) . This product is to be add for remaining number [ 37 + (-16) = 21].

Here 21 is divisible by 7 so the given number is also divisible by 7

#### Divisibility rules for 11 :

A number is divisible by 11, if the **difference of the the sum of the digits in the odd places and the sum of the digits in the even places is zero or is divisible by 11.**

**Examples :**

1.Check the divisibility of a number 2846767 by 11.

- Odd places of the given number = 7 + 7 + 4 + 2 = 20
- Even places of the given number = 6+6+8 = 20
- The difference of the sum of odd and even places is ” 0″ . So the number
**2846767 is divisible by 11.**

2.Check the divisibility of a number 964712435 by 11.

- Odd places of the given number = 5 + 4 + 1 +4 +9 = 23
- Even places of the given number = 3 + 2 + 7 + 6 = 18
- The difference of the sum of odd and even places is ” 5″ . So the number
**964712435 is not divisible by 11.**

#### Divisibility rule for 13 :

We use **oscillator (+4)** for divisibility test of 13.

for example take number 7631

step 1 : Separate unit place from the given number and multiplied by ” 4″ for unit digit place ( 1 x 4 = 4) . This product is to be add for remaining number [ 763 + (4) = 767 ].

step 2 : Separate unit place from 763 and multiplied by “4” for unit digit place ( 7 x 4 = 28) . This product is to be add for remaining number [ 76 + (28) = 104].

Here 104 is divisible by 13 so the given number 7631 is also divisible by 13. ( If we continue the same process for next step than final value will get 13)

#### Divisibility rule for 17 :

We use **oscillator (-5)** for divisibility test of 17.

for example take number 1525665

step 1 : Separate unit place from the given number and multiplied by ” -5″ for unit digit place ( -5 x 5 = -25) . This product is to be add for remaining number [ 152566 + (-25) = 152541].

step 2 : Separate unit place from 152541 and multiplied by “-5” for unit digit place ( -5 x 1 = -5) . This product is to be add for remaining number [ 15254 + (-5) = 15249 ].

step 3 : Separate unit place from 16249 and multiplied by “-5” for unit digit place ( -5 x 9 = -45) . This product is to be add for remaining number [ 1524 + (-45) = 1479].

step 4 : Separate unit place from 1579 and multiplied by “-5” for unit digit place ( -5 x 9 = -45) . This product is to be add for remaining number [ 147 + (-45) = 102].

Here 102 is divisible by 17 so the given number is also divisible by 17. ( If we continue the same process for next step than final value will get 0)

#### Divisibility rule for 19 :

We use **oscillator (+2)** for divisibility test of 19.

for example take number 112043

step 1 : Separate unit place from the given number and multiplied by ” 2″ for unit digit place ( 2 x 3 = 6) . This product is to be add for remaining number [ 11204 + (6) = 11210 ].

step 2 : Separate unit place from 11210 and multiplied by “2” for unit digit place ( 2 x 0 = 0) . This product is to be add for remaining number [ 1121 + (0) = 1121].

step 3 : Separate unit place from 1121 and multiplied by “2” for unit digit place ( 2 x 1 = 2) . This product is to be add for remaining number [ 112 + (2) = 114].

Here 114 is divisible by 19 so the given number is also divisible by 19. ( If we continue the same process for next step than final value will get 19)

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### Number System:

Formulas for Sum of n Consecutive numbers

GCD and LCM Problems & Solutions

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