Divisibility Rule (Divisibility Test - 11 to 20)

In many competitive questions, it becomes very important to find whether given number is divisible by another given number or not. It is required when we solve HCF (Highest Common Factor), LCM (Least Common Multiple), Composite numbers, Co-Prime numbers, multiples, factors, etc.

So this topic is extensively important in understanding many parts of number mathematics and algebra as well because all operations in number mathematics are always true for algebra and other mathematics also.

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Divisibility Rule (Divisibility Test - 11 to 20)

Divisibility test means testing of a number whether it is divisible completely by another given number. It is always means that the first number (Dividend) and the second number (Divisor) are integers. Result of a division is called Quotient and if something is left, it is called a remainder.

For Example:

456382 ÷ 12 = (Q=38031 and R=10)

Important:

"If in a division, quotient is an integer, and remainder is zero, it means that dividend is completely divisible by divisor."

"If in a division, either quotient is a fraction with remainder zero or quotient is integer with remainder non-zero, it means that dividend is not completely divisible by divisor."

Click Here for Numbers: Divisibility Rule (1-10)

11. Divisible By 11: If difference of sums of alternative digits of a given number is divisible by 11, then the given number is divisible by 11.

If difference of the sum of odd digits and the sum of even digits from a given number is divisible by 11, then the given number is divisible by 11.

First we will understand alternate places or even and odd places in a given number

(Figure: Alternative Digits in a Number)

Step for Test: 1. Sum all odd alternate digits from the given number. Say it Odd Alternative Sum  2. sum all even alternate digits from the given number. Say it Even Alternative Sum 3. Subtract Even Alternative Sum from Odd Alternative Sum. Say it subtracted result. 4. If Subtracted Result is divisible by 11, then the given number is divisible by 11.

For Example:

Let us check whether 9182701 is divisible by 11.

First sum alternative odd digits = 9 + 8 + 7 + 1 = 25

Then Sum alternative even digits = 1 + 2 + 0 = 3

difference of two results = 25 -3 = 22

We can see that 22 is divisible by 11.

So the given number 9182701 is divisible by 11.

Verification: Now divide the number by 11 using conventional division method, we get that the given number 9182701 is divisible by 11, and quotient is 834791 along ‬with remainder 0.

12. Divisible By 12: If a number is divisible by 4 as well as divisible by 3, then the number is completely divisible by 12.

So it is very clear that the number should follow both the rules of divisibility test by 4 and divisibility test by 3 simultaneously.

Note: If number is not divisible by any one of 4 and 3, then number will not be divisible by 12. If number is divisible by 4 by not divisible by 3, it is not divisible by 12. Similarly, if number is not divisible by 2 or 4 but divisible by 3, it is not divisible by 12.

Steps for Test: 1. Divisibility Test by 4:       First check whether last place of the number is 0,2,4,6,8 means number is even.      If number is even, then write last two digits (10's place and 1's place) as a number from the given number.      If this number is divisible by 4, then the given number is divisible by 4.     If this number is not divisible by 4, then the given number is not divisible by 4 2. If Number is divisible by 4, then we proceed for step 3. 3. Test for Divisibility by 3      Sum all the digits      If sum found is more than one digit number, then again sum all the digits of the previous sum.     Repeat the process until we get sum of digits as a one digit number.     If final sum is divisible by 3, then the given number is divisible by 3.     If final sum is not divisible by 3, then the given number is not divisible by 3.  4. If number is divisible by 4 and 3, The given number is divisible by 12.

For Example:

Example 1:  

Let us check whether 74558232 is divisible by 12.

Step 1: Last digit is 2 so the given number is even.

Step 2: Divisibility Test By 4:

   Write last two digits as a number i.e. 32

   We can see that 32 is divisible by 4,                                  (Divisible By 4)

Step 3: Divisibility Test By 3: 

Sum of All digits = 7 + 4 + 5 + 5 + 8 + 2 + 3 + 2 = 36

Again Sum of all digits = 3 + 6 = 9                                         (Divisible by 3)

So the number is divisible by 3.

Since the given number is divisible by 4 and 3, Hence the given number 74558232 is divisible by 12.

Verification: Now divide the number by 12 using conventional division method, we get that the given number 74558232 is divisible by 6 and quotient is 6213186 ‬with remainder 0.

Example 2: 

Let us take another example: Check whether 32644238 is divisible by 12?

Step 1: since last digit of the number is 8 hence given number is an even number.

Step 2: Divisibility Test By 4: 

   Write last two digits as a number i.e. 38

   We can see that 38 is not divisible by 4, So number is not divisible by 4.

Since the given number is not divisible by 4 Hence the given number 32644238 is not divisible by 12.

Verification: Now divide the number by 12 using conventional division method, we get that the given number 32644238 is not divisible by 12 and quotient 2720353 ‬with remainder 2 (≠ 0).

Example 3: 

Let us take another example: Check whether 4876223 is divisible by 12?

Step 1: Divisibility Test By 2: 

Since last digit of number is 3 which is not divisible by 2, so number is not divisible by 2 or not even number.

Since the given number is not divisible by 2, Hence the given number 4876223 is not divisible by 12.

Verification: Now divide the number by 12 using conventional division method, we get that the given number 4876223  is not divisible by 12 and quotient is 406351‬ with remainder 11 (≠ 0).

13. Divisible By 13: A number is divisible by 13 if result obtained from the addition of four times the last digit (one's place) from the remaining part of the number is divisible by 13.

Steps for Test: 1. Write the last digit of number i.e. one's place of the number. 2. Multiply it by 4. 3. Add this number to the remaining portion of the number. We will obtain a result. 4. Repeat above three steps for the result obtained many times until we get a number less than 130. 5. If final result is divisible by 13, then the given number is divisible by 13.

For Example:

Let us check whether 262743 is divisible by 13.

The unit place of this number is 3

Multiply it by 4 = 3 × 4 = 12

Remaining Portion of number =         26274

add the number multiplied by 4 =       26274  +   12  =  26286

Repeat the process: The unit place of new result is 6

Multiply it by 4 = 6 × 4 = 24

add this number in remaining portion =   2628  +   24  =  2652

Repeat the process: The unit place of new result is 2

Multiply it by 4 = 2 × 4 = 8

add this number in remaining portion =   265  +   8  =  273

Repeat the process: The unit place of new result is 3

Multiply it by 4 = 3 × 4 = 12

add this number in remaining portion =   27  +   12  =  39              (Divisible By 13)

So the given number 262743 is  divisible by 13.

Verification: Now divide the number by 13 using conventional division method, we get that the given number 262743 is  divisible by 13 and quotient is 20211 ‬with remainder 0.

14. Divisible By 14: If a number is divisible by 2 as well as divisible by 7, then the number is completely divisible by 14. 

So it is very clear that the number should follow both the rules of divisibility test by 2 and divisibility test by 7 simultaneously.

Note: If number is not divisible by any one of 2 and 7, then number will not be divisible by 14. If number is divisible by 2 by not divisible by 7, it is not divisible by 14. Similarly, if number is not divisible by 2 but divisible by 7, it is not divisible by 14.

Steps for Test: 1. Check whether last digit of given number is 0,2,4,6,8    If Yes then the given number is divisible by 2 then we proceed for next step.   If No then the given number is not divisible by 2 then not need to proceed further, the number is not divisible by 14 as well. 2. If Number is divisible by 2, then we proceed towards step 3. 3. Write the last digit of number i.e. one's place of the number. 4. Double it (Means multiply it by 2) 5. Subtract this number from the remaining portion of the number. We will obtain a result. 6. Repeat above three steps for the result obtained many times until we get a number less than 70. 7. If final result is divisible by 7, then the given number is divisible by 14.     If final result is not divisible by 7, then the given number is not divisible by 14.

For Example:

Let us check whether 98994 is divisible by 14.

The unit place of this number is 4, so this is an even number.          (Divisible by 2)

Doubling 4, we get 4 × 2 = 8

Remaining Portion of number = 9899

Subtract double number            9899  -  8  =  9891

Repeat the process: The unit place of new result is 1

Doubling 1, we get 2

Subtract it from new remaining portion      989  -  2  =  987

Repeat the process: The unit place of new result is 7

Doubling 7, we get 14

Subtract it from new remaining portion      98  -  14  =  84

Repeat the process: The unit place of new result is 4

Doubling 4, we get 8

Subtract it from new remaining portion      8  -  4  =  0                        (Divisible By 7)

So the given number 98994 is  divisible by 7.

Since the given number is even as well as it is divisible by 7, so the given number is divisible by 14.

Verification: Now divide the number by 14 using conventional division method, we get that the given number 98994 is divisible by 14 and quotient is 7071 ‬with remainder 0.

15. Divisible By 15: If the given number is divisible by 5 and 3 both, then the number is divisible by 15.

So it is very clear that the number should follow both the rules of divisibility test by 5 and divisibility test by 3 simultaneously.

Note: If number is not divisible by only any one of 5 and 3, then number will not be divisible by 15. If number is divisible by 5 by not divisible by 3, it is not divisible by 15. Similarly, if number is not divisible by 5 but divisible by 3, it is not divisible by 15.

Step for Test: 1. If unit place of the given number is 0, 5, then it is divisible by 5.    If yes, proceed for the next step,    If no, then number is not divisible by 15. 2. If number is divisible by 5, we will proceed for divisibility test by 3. 3. Sum all the digits 4. If sum found is more than one digit number, then again sum all the digits of the previous sum. 5. Repeat the process until we get sum of digits as a one digit number. 6. If final sum is divisible by 3, then the given number is divisible by 15.     If final sum is not divisible by 3, then the given number is not divisible by 15.

For Example:

Example 1: 

Let us check whether 162738915 is divisible by 15.

Since last digit is 5 hence number is divisible by 5.                        (Divisible by 5)

Sum of All digits = 1 + 6 + 2 + 7 + 3 + 8 + 9 + 1 + 5 = 42

Again Sum all digits = 4 + 2 = 6                                                      (Divisible by 3)

We can see that the sum of all digits in the given number is divisible by 3, so the given number 162738915 is divisible by 3.

Since the given number is divisible by 5 as well as 3, So, the given number is divisible by 15.

Verification: Now divide the number by 15 using conventional division method, we get that the given number 162738915 is divisible by 15 and quotient is 10849261 along with remainder 0.

Example 2: 

Now let us take another example: Check whether 7635428740 is divisible by 3?

Since one's digit is 0, Number is divisible by 5.                            (Divisible By 5)

Sum of all digits = 7 + 6 + 3 + 5 + 4 + 2 + 8 + 7 + 4 + 0 = 46

Again sum all digits = 4 + 6 = 10                                                  (Not divisible by 3)

We can see that sum of all digits in the given number is not divisible by 3, so the given number 7635428743 is not divisible by 3 and hence 15.

Verification: Now divide the number by 15 using conventional division method, we get that the given number 7635428743 is not divisible by 15 and quotient is 509028582 with remainder 10 (≠ 0).

16. Divisible By 16: If combination of last Four digits (Thousand's, Hundred's, Ten's and one's Place) as a number is divisible by 16, then the given number is divisible by 16.

Steps for Test: 1. First check whether last place of the number is 0,2,4,6,8 means number is even. 2. If number is even then write last Four digits (1000's place, 100's place, 10's place and 1's place) as a number from the given number. 3. If this number is divisible by 16, then the given number is divisible by 16.     If this number is not divisible by 16, then the given number is not divisible by 16.

For Example:

Let us check whether 9753120 is divisible by 16.

Checking last digit of the number is 0. It means number is even.

Write last Four digit number from the given number i.e. 3120.

If we divide 3120 by 16, then  we get quotient 195 along with remainder 0.

So 3120 is divisible by 16

So the given number 9753120 is divisible by 16.

Verification: Now divide the number by 16 using conventional division method, we get that the given number 9753120 is divisible by 8 and quotient is 609570 along ‬with remainder 0.

17. Divisible By 17: A number is divisible by 17 if result obtained from the subtraction of five times the last digit (one's place) from the remaining part of the number is divisible by 17.

Steps for Test: 1. Write the last digit of number i.e. one's place of the number. 2. Multiply it by 5. 3. Subtract this number from the remaining portion of the number. We will obtain a result. 4. Repeat above three steps for the result obtained many times until we get a number less than 170. 5. If final result is divisible by 17, then the given number is divisible by 17.

For Example:

Let us check whether 27897 is divisible by 17.

The unit place of this number is 7

Five times the 7, we get 7 × 5 = 35

Remaining Portion of number = 2789

Subtract double number            2789  -  35  =  2754

Repeat the process: The unit place of new result is 4

Five times the 4, we get 4 × 5 = 20

Subtract it from new remaining portion     275  -  20  =  255

Repeat the process: The unit place of new result is 5

Five times the 5, we get 5 × 5 = 25

Subtract it from new remaining portion     25  -  25  =  0         (Divisible By 17)

So the given number 27897 is  divisible by 17.

Verification: Now divide the number by 17 using conventional division method, we get that the given number 27897 is divisible by 17 and quotient is 1641 ‬along with remainder 0.

18. Divisible By 18: If a number is divisible by 2 as well as divisible by 9, then the number is completely divisible by 18.

So it is very clear that the number should follow both the rules of divisibility test by 2 and divisibility test by 9 simultaneously.

Note: If number is not divisible by only any one of 2 and 9, then number will not be divisible by 18. If number is divisible by 2 by not divisible by 9, it is not divisible by 18. Similarly, if number is not divisible by 2 but divisible by 9, it is not divisible by 18.

Step for Test: 1. Test for divisibility by 2.       If last digit of number is 0,2,4,6,8 or number is even then it is divisible by 2. Then we proceed for next step      If last digit of number is neither of 0,2,4,6,8 or number is odd, then it is not divisible by 2. Then stop testing and declare that number is not divisible by 18. 2. Test for divisibility by 9      Sum all the digits      If sum found is more than one digit number, then again sum all the digits of the previous sum.      Repeat the process until we get sum of digits as a one digit number or you can stop up to two digit number less than 90.      If final sum is divisible by 9, then the given number is divisible by 18.     If final sum is not divisible by 9, then the given number is not divisible by 18.

For Example:

Let us check whether 162738954 is divisible by 9.

The unit place has 4 it means number is divisible by 2.

Sum of All digits = 1 + 6 + 2 + 7 + 3 + 8 + 9 + 5 + 4 = 45         (Divisible by 9)

Again Sum all digits = 4 + 5 = 9                                                (Divisible by 9)

We can see that the sum of all digits in the given number is divisible by 9, so the given number 162738954 is divisible by 9.

Since the given number is divisible by 2 as well as 9, so the given number 162738954 is divisible by 18.

Verification: Now divide the number by 18 using conventional division method, we get that the given number 162738954 is divisible by 18 and quotient is 9041053 along with remainder 0.

19. Divisible By 19: A number is divisible by 19 if result obtained from the addition of double of last digit (one's place) with the remaining part of the number is divisible by 19.

Steps for Test: 1. Write the last digit of number i.e. one's place of the number. 2. Double it (Means multiply it by 2) 3. Add this number from the remaining portion of the number. We will obtain a result. 4. Repeat above three steps for the result obtained many times until we get a number less than 190. 5. If final result is divisible by 19, then the given number is divisible by 19.

For Example:

Let us check whether 27968 is divisible by 19.

The unit place of this number is 8

Doubling 4, we get 8 × 2 = 16

Remaining Portion of number = 2796

adding double number               2796  +  16  =  2812

Repeat the process: The unit place of new result is 2

Doubling 2, we get 4

Adding it from new remaining portion      281  +   4  =  285

Repeat the process: The unit place of new result is 5

Doubling 5, we get 10

adding it from new remaining portion      28  +  10  =  38        (Divisible By 19)

So the given number 27968 is  divisible by 19.

Verification: Now divide the number by 19 using conventional division method, we get that the given number 27968 is divisible by 19 and quotient is 1472 ‬with remainder 0.

20. Divisible By 20: a number is divisible by 20 if combination of last two digits (Ten's Place and One's Place) as a number is divisible by 20.

Steps for Test: Write combination of last two digits (10's place and 1's place) as a number If number is divisible by 20 then the given number is divisible by 20.

For Example:

Let us check  whether 3585940 is divisible by 20.

Last two digits as a number = 40      (Divisible by 20)

So the given number 3585940 is divisible by 20.

Verification: Now divide the number by 20 using conventional division method, we get that the given number 3585940 is divisible by 20 and quotient is 179297 along ‬with remainder 0.

Click Here for Numbers: Divisibility Rule (1-10)

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