TESTS OF DIVISIBILITY 

DIVISIBILITY TESTS BELOW 50 

DIVISIBILITY TEST WITH 2 :- A number is divisible by 2 if its last digit or digit in units place is 0,2,4,6,8. 

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DIVISIBILITY TEST WITH 3 :- A number is divisible by 3 if the sum of its digits is also divisible by 3.

               Example :- 624 : 6 + 2 + 4 = 12 

                                      → 12 : 1 + 2 = 3 

                               so 624 is divisible by 3. 

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DIVISIBILITY TEST WITH  5 :- A number is divisible by 5 if its last digit or digit in units place is 0 or 5. 

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DIVISIBILITY TEST WITH  6 :- A number is divisible by 6 if it is even and divisible by 3. (or) if it is divisible by both 2 and 3.

                            Example :- 624 : a) It is an even number.

                                                  b) 624 : 6 + 2 + 4 = 12 

                                                     → 12 : 1 + 2 = 3 

                                                 so 624 is divisible by 3.

                                    Hence we can say 624 is divisible by 6.                                     

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DIVISIBILITY TEST WITH  7 :- Double  the last digit and subtract it from the remaining leading truncated number. If the result is divisible by 7, then so was the original number. Apply this rule over and over again as necessary.

                                Example :- 714 : 71 - 2 x 4 = 71 - 8 = 63.

                                                      63:   63 = 9 x 7. 

                                               So 714 is divisible by 7.

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DIVISIBILITY TEST WITH  8 :- A number is divisible by 8 if  the number formed by its digits in hundred's, ten's and units place is divisible by 8.

      Example :- 19624 : digit in hundred's place → 6 

                                        digit in ten's place → 2 

                                        digit in unit's place → 4 

                                          so number formed → 624 which is divisible by 8. 

                                     Hence we can say 19624 is divisible by 8.  

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DIVISIBILITY TEST WITH 9 :-  A number is divisible by 9 if the sum of its digits is also divisible by 9.

                            Example :- 693 : 6 + 9 + 3 = 18 

                                                          → 18 : 9 x 2 

                                              so 693 is divisible by 9.

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DIVISIBILITY TEST WITH  10 :- A number is divisible by 10 if its last digit or digit in units place is 0. 

                             Example :- 3250,200,145670, etc 

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DIVISIBILITY TEST WITH 11 :-  Subtract the last digit from the remaining leading truncated number. If the result is divisible by 11, then so was the first number. Apply this rule over and over again as necessary.

                Example :- 19151 : 1915 - 1 = 1914 

                                    → 1914 : 191 - 4 = 187 

                                    → 18 - 7 = 11 

                                   So 19151 is divisible by 11.

                                                                (OR)

  If the difference of the sum of its digits in odd places and in even places (starting from the units's place) is divisible by 11.

                                   Example :- 19151 : Sum of Odd places digits = 1 + 1 + 1 = 3

                                                           → Sum of Even places digits = 5 + 9 = 14 

                                                           → difference = 14 - 3 = 11 which is divisible by 11. 

                                                               So 19151 is divisible by 11.  

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DIVISIBILITY TEST WITH 13 :-  Add four times the last digit to the remaining leading truncated number. If the result is divisible by 13, then so was the first number. Apply this rule over and over again as necessary.

                         Example :- 50661 : 5066 + 4 x 1 = 5066 + 4 = 5070 

                                               → 5070 : 507 + 4 x 0 = 507 + 0 = 507 

                                               → 507 : 50 + 4 x 7 = 50 + 28 = 78 

                                               → 78 : 7 + 4 x 8 = 7 + 32 = 39 

                                               → 39 : 3 x 13 

                                            So 50661 is divisible by 13.

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DIVISIBILITY TEST WITH 17 :- Subtract five times the last digit from the remaining leading truncated number. If the result is divisible by 17, then so was the first number. Apply this rule over and over again as necessary.

                          Example :- 3978 : 397 - 5 x 8 = 397 - 40 = 357 

                                            → 357 : 35 - 5 x 7 = 35 - 35 = 0  (or) 357 = 21 x 17 

                                                 So 3978 is divisible by 17. 

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DIVISIBILITY TEST WITH 19 :-  Add two times the last digit to the remaining leading truncated number. If the result is divisible by 19, then so was the first number. Apply this rule over and over again as necessary.

                         Example :- 101156 : 10115 + 2 x 6 = 10115 + 12 = 10127 

                                                → 10127 : 1012 + 2 x 7 = 1012 + 14 = 1026 

                                                → 1026 : 102 + 2 x 6 = 102 + 12 = 114 

                                                → 114 : 11 + 2 x 4 = 11 + 8 = 19. 

                                                     So 101156 is divisible by 19.   

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DIVISIBILITY TEST WITH 23 :-  Add seven times the last digit to the remaining leading truncated number. If the result is divisible by 23, then so was the first number. Apply this rule over and over again as necessary.

                  Example :- 101200 : 10120 + 7 x 0 = 10120 + 0 = 10120 

                                       → 10120 : 1012 + 7 x 0 = 1012 + 0 = 1012 

                                      → 1012 : 101 + 7 x 2 = 101 + 14 = 115 

                                      → 115 : 11 + 7 x 5 = 11 + 35 = 46 

                                      → 46 : 46 = 2 x 23 . 

                                         So 101200 is divisible by 23.   

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DIVISIBILITY TEST WITH 29 :- Add three times the last digit to the remaining leading truncated number. If the result is divisible by 29, then so was the first number. Apply this rule over and over again as necessary.

                Eample :- 101181 : 10118 + 3 x 1 = 10118 + 3 = 10121 

                                    → 10121 : 1012 + 3 x 1 = 1012 + 3 = 1015 

                                    → 1015 : 101 + 3 x 5 = 101 + 15 = 116 

                                      → 116 : 11 + 3 x 6 = 11 + 18 = 29 

                                         So 101181 is divisible by 29.     ========================================================================================

DIVISIBILITY TEST WITH 31 :- Subtract three times the last digit from the remaining leading truncated number. If the result is divisible by 31, then so was the first number. Apply this rule over and over again as necessary.

                    Example :- 10075 : 1007 - 3 x 5 = 1007 - 15 = 992 

                                                  → 992 : 99 - 3 x 2 = 99 - 6 = 93

                                                     → 93 : 93 = 3 x 31 

                                              So 10075 is divisible by 31. 

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DIVISIBILITY TEST WITH 37 :- Subtract eleven times the last digit from the remaining leading truncated number. If the result is divisible by 37, then so was the first number. Apply this rule over and over again as necessary.

                       Example :-  1006437 : 100643 - 11 x 7 =  100643 - 77 = 100566 

                                              →100566 : 10056 - 11 x 6 = 10056 - 66 = 9990 

                                              → 9990 : 999 - 11 x 0 = 999 - 0 = 999 

                                              → 999 : 99 - 11 x 9 = 99 - 99 = 0 . (or) 999 = 37 x 27 

                                                       So 1006437 is divisible by 37.

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DIVISIBILITY TEST WITH 41 :-  Subtract four times the last digit from the remaining leading truncated number. If the result is divisible by 41, then so was the first number. Apply this rule over and over again as necessary.

                     Example :- 1006427 : 100642 - 4 x 7 = 100642 - 28 = 100614 

                                                 → 100614 : 10061 - 4 x 4 = 10061 - 16 = 10045 

                                                 → 10045 : 1004 - 4 x 5 = 1004 - 20 = 984 

                                                  → 984 : 98 - 4 x 4 = 98 - 16 = 82 

                                                       → 82 : 41 x 2. 

                                                    So 1006427 is divisible by 41.

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DIVISIBILITY TEST WITH 43 :-  Add thirteen times the last digit to the remaining leading truncated number. If the result is divisible by 43, then so was the first number. Apply this rule over and over again as necessary.

     Example :- 1006415 : 100641 + 13 x 5 = 100641 + 65 = 100706 

                    → 100706 : 10070 + 13 x 6 = 10070 + 78 = 10148 

                    → 10148 : 1014 + 13 x 8 = 1014 + 104 = 1118 

                    → 1118 : 111 + 13 x 8 = 111 + 104 = 215 

                    → 215 : 21 + 13 x 5 = 21 + 65 = 86 

                    → 86 : 86 = 43 x 2 . 

                        So 1006415 is divisible by 43.

                                                         (OR)

Subtract thirty times the last digit from the remaining leading truncated number. If the result is divisible by 43, then so was the first number. Apply this rule over and over again as necessary.

     Example :- 1006415 : 100641 - 30 x 5 =  100641 - 150 = 100491 

                       → 100491 : 10049 - 30 x 1 = 10049 - 30 = 10019 

                       → 10019 : 1001 - 30 x 9 = 1001 - 270 = 731 

                       → 731 : 73 - 30 x 1 = 73 -  30 = 43 . 

                       So 1006415 is divisible by 43.

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DIVISIBILITY TEST WITH 47 :-  Subtract fourteen times the last digit from the remaining leading truncated number. If the result is divisible by 47, then so was the first number. Apply this rule over and over again as necessary.

     Example :- 107066 : 10706 - 14 x 6 =  10706 - 84 = 10622 

                         → 10622 : 1062 - 14 x 2 =  1062 - 28 = 1034 

                         → 1034 : 103 - 14 x 4 = 103 - 56 = 47 . 

                      So 107066 is divisible by 47.

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