Divisibility Rule By Prime Numbers (2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37)

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.

Divisibility Rule (Divisibility Test - 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37) By Prime Numbers

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."

1. Divisible By 2: An integer having 0, 2, 4, 6, 8 at its unity place or Last digit, is divisible by 2.

Steps of Test: There is only one step of test i.e. check the last digit is 0,2,4,6,8 or number is even number

For Example: 

2340, 12, 34, 56, 35798, etc

Note: The integers which are completely divisible by 2 are called even numbers.

2. Divisible By 3: If sum of all digits in an integer is divisible by 3, then the number is divisible by 3.

Step for Test: 1. Sum all the digits 2. If sum found is more than one digit number, then again sum all the digits of the previous sum. 3. Repeat the process until we get sum of digits as a one digit number. 4. 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.

For Example:

Example 1: Let us check whether 162738951 is divisible by 3.

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

                           = 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 162738951 is divisible by 3.

Verification: Now divide the number by 3 using conventional division method, we get that the given number 162738951 is divisible by 3 and quotient is 54246317‬ with remainder 0.

Example 2: Now let us take another example: Check whether 7635428743 is divisible by 3?

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

                          = 49

Again sum all digits = 4 + 9 = 13           (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.

Verification: Now divide the number by 4 using conventional division method, we get that the given number 7635428743 is not divisible by 3 and quotient is 2545142914 with remainder 1 (≠ 0).

3. Divisible By 5: If Unit place (Last place) of number has either 0 or 5 then number is divisible by 5.

Steps of Test: There is only one step of test i.e. check the last digit is 0, 5.

For Example: 7520, 23456785, 975315, etc.

4. Divisible By 7: A number is divisible by 7 if result obtained from the subtraction of double of last digit (one's place) from the remaining part of the number is divisible by 7.

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. 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 70. 5. If final result is divisible by 7, then the given number is divisible by 7.

For Example:

Example 1: Let us check whether 98994 is divisible by 7.

The unit place of this number is 4

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.

Verification: Now divide the number by 7 using conventional division method, we get that the given number 98994 is divisible by 7 and quotient is 14142 ‬with remainder 0.

Example 2: Let us check whether 2268 is divisible by 7.

The unit place of this number is 8

Doubling 8, we get 8 × 2 = 16

Remaining Portion of number = 226

Subtract double number            226  -  16  =  210

Repeat the process: The unit place of new result is 0

Doubling 0, we get 0

Subtract it from new remaining portion      21  -  0  =  21        (Divisible By 7)

So the given number 2268 is  divisible by 7.

Verification: Now divide the number by 7 using conventional division method, we get that the given number 2268 is divisible by 7 and quotient is 324 ‬with remainder 0.

5. 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.

6. 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.

7. 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.

8. 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.

9. Divisible By 23: There are two rules for divisibility by 23. Whichever you get easy, you should apply in exams

Rule A: A number is divisible by 23 if result obtained from the addition of seven times the last digit (one's place) to the remaining part of the number is divisible by 23.

Rule B: A number is divisible by 23 if result obtained from the addition of three times the last two digits (combination of ten's place and one's place) to the remaining part of the number is divisible by 23.

Rule A: Steps for Test: 1. Write the last digit of number i.e. one's place of the number. 2. Multiply it by 7 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 in two digits. 5. If final result is divisible by 23, then the given number is divisible by 23.  Rule B: Steps for Test: 1. Write the last two digits of number i.e. combination of ten's and one's place of the number. 2. Multiply it by 3 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 230. 5. If final result is divisible by 23, then the given number is divisible by 23.

For Example:

Let us check whether 93173 is divisible by 23.

Rule A: First Write last digit of the number i.e. 3

              Multiply it by 7 = 3 × 7 = 21

              Add this number in remaining portion = 9317 + 21 = 9338

              Repeat above steps

              Last digit of the new result = 8

              Multiply it by 7 = 8 × 7 = 56

              Add this number in remaining portion = 933 + 56 = 989

              Repeat above steps

              Last digit of the new result = 9

              Multiply it by 7 = 9 × 7 = 63

              Add this number in remaining portion = 98 + 63 = 161

              Repeat above steps

              Last digit of the new result = 1

              Multiply it by 7 = 1 × 7 = 7

              Add this number in remaining portion = 16 +7 = 23                   (Divisible By 23)

              Hence the given number 93173 is divisible by 23.

Rule B: First Write last two digits of the number i.e. 73

              Multiply it by 3 = 73 × 3 = 219

              Add this number in remaining portion = 931 + 219 = 1150

              Repeat above steps

              Write last two digits of the new number i.e. 50

              Multiply it by 3 = 50 × 3 = 150

              Add this number in remaining portion = 11 + 150 = 161            (Divisible By 23)

              Hence the given number 93173 is divisible by 23.

Verification: Now divide the number by 23 using conventional division method, we get that the given number 93173 is divisible by 23, and quotient is 4051 along ‬with remainder 0.

10. Divisible By 29: A number is divisible by 29 if result obtained from the addition of three times the last digit (one's place) from the remaining part of the number is divisible by 29.

Steps for Test: 1. Write the last digit of number i.e. one's place of the number. 2. Multiply it by 3. 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 290. 5. If final result is divisible by 29, then the given number is divisible by 29.

For Example:

Let us check whether 61857 is divisible by 29.

The unit place of this number is 7

Multiply it by 3 = 7 × 3 = 21

Remaining Portion of number =         6185

add the number multiplied by 4 =       6185 + 21 = 6206

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

Multiply it by 3 = 6 × 3 = 18

add this number in remaining portion =   620 + 18 = 638

Repeat the process: The unit place of new result is 8

Multiply it by 3 = 8 × 3 = 24

add this number in remaining portion =   63  +   24  =  87              (Divisible By 29)

So the given number 61857 is  divisible by 29.

Verification: Now divide the number by 29 using conventional division method, we get that the given number 61857 is  divisible by 29 and quotient is 2133 ‬with remainder 0.

11. Divisible By 31: A number is divisible by 31 if result obtained from the subtraction of three times the last digit (one's place) from the remaining part of the number is divisible by 31.

Steps for Test: 1. Write the last digit of number i.e. one's place of the number. 2. Multiply it by 3. 3. Subtraction 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 310. 5. If final result is divisible by 29, then the given number is divisible by 31.

For Example:

Let us check whether 55149 is divisible by 31.

The unit place of this number is 9

Multiply it by 3 = 9 × 3 = 27

Remaining Portion of number =         5514

add the number multiplied by 4 =       5514 - 27 = 5487

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

Multiply it by 3 = 7 × 3 = 21

add this number in remaining portion =   548 - 21 = 527

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

Multiply it by 3 = 7 × 3 = 21

add this number in remaining portion =   52  -   21  =  31              (Divisible By 31)

So the given number 55149 is  divisible by 31.

Verification: Now divide the number by 31 using conventional division method, we get that the given number 55149 is  divisible by 31 and quotient is 1779 ‬with remainder 0.

12. Divisible By 37: There are two rules for divisibility by 23. Whichever you get easy, you should apply in exams

Rule A: A number is divisible by 37 if result obtained from the subtraction of eleven times the last digit (one's place) to the remaining part of the number is divisible by 37.

Rule B: A number is divisible by 37 if result obtained from the addition of numbers by taking sets of three digits from right to left without reforming three digit numbers is divisible by 37.

Rule A: Steps for Test: 1. Write the last digit of number i.e. one's place of the number. 2. Multiply it by 11 3. Subtract 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 in two digits. 5. If final result is divisible by 37, then the given number is divisible by 37.  Rule B: Steps for Test: 1. Write all the numbers of three digits taking from right to left from the given number 2. Add all the numbers 3. Repeat above two steps for the result obtained many times until we get a number less than 1000. 4. If final result is divisible by 37, then the given number is divisible by 37.

For Example:

Let us check whether 310837 is divisible by 37.

Rule A: First Write last digit of the number i.e. 7

              Multiply it by 11 = 7 × 11 = 77

              Add this number in remaining portion = 31083 - 77  = 31006

              Repeat above steps

              Last digit of the new result = 6

              Multiply it by 11 = 6 × 11 = 66

              Add this number in remaining portion = 3100 - 66 = 3034

              Repeat above steps

              Last digit of the new result = 4

              Multiply it by 11 = 4 × 11 = 44

              Add this number in remaining portion = 303 - 44 = 259            (Divisible By 37)

              Hence the given number 310837 is divisible by 37.

Rule B: Write 310837 as sets of three digits numbers 310,837

              Adding all sets 310 + 837 = 1147

              Repeat above steps: Write 1147 as sets of three digits numbers 1,147

              Adding all sets 1 + 147 = 148                                                  (Divisible By 37)

              Hence the given number 310837 is divisible by 37.

Verification: Now divide the number by 37 using conventional division method, we get that the given number 310837 is divisible by 37, and quotient is 8401 along ‬with remainder 0.