Division by 11

                     Here we learn division by 11 It is very easy and you can do it mentally very fast

                             First  we check particular number is divisible by 11 or not  We  already  discuss  this  method  in
                             lesson 11 .There are only two possibilities either that number is divisible by 11 or it is not divisible by 11
                                Number is divisible by 11      Let the number is 47894
                                                                                 4  7  8  9  4   
                                                                                 total of odd position -total of even position
                                                                                (4+8+4)-(7+9)
                                                                                 16 -16 =0 so this number is divisible by 11
                                   Steps                                     Write  number  47894  ,take last digit ie 4
                                                                                 Write it extreme right .............4. Now subtract 4 from
                                                                                  second last digit ie9-4=5 write before 4 our partial answer is
                                                                                  .............54 .Again subtract 5from8          8-5=3  Write before
                                                                                   ......54 .Now our partial answer is ....354 ,Repeat these steps
                                                                                    up to first digit and get required answer  .Here we subtract 3
                                                                                   from7 , 7-3=4    Our partial answer is 4354 Again subtract 4
                                                                                   from4       4-4=0  0 is indication we are on right track  So 4354
                                                                                    is required answer

                                                                                   4              7           8              9               4
                                                                                   4-4          7-3        8-5          9-4

                                                                                                                                                    
                                                                                    0                4           3              5              4   start from here

                                                                                                              =04354                        

                                           Let  us see one more example Here we also consider carry

                                              5225=
                                                 11

                                                  5              2             2              5
                                                4-4         11-7         12-5           |
                                                    |              |               |              |.
                                                   0             4              7             5         start  from here

                                                  = 0475  is our  required answer

                           Number is not  divisible by 11    Same method as we  discuss above but we make first
                                                                                 number  divisible by 11. Let us take a number  7539
                                                                                 This number is not divisible by 11 because difference
                                                                                 of total of odd  and even position is not 0 or 11 .22.....

                                                                                    7  5  3 9      (7+3)-(5+9) = 10 -14 =--4

                                                                                  If  we subtract 4 from this number ,now this number is certainly
                                                                                  divisible by 11 . And that number is 7539-4 =7535  By this
                                                                                   way we find remainder .Here our remainder is 4

                                                                                7539       =      7535     +    4
                                                                                   11                 11             11

                                                                            
                                                                                                   Now we can solve first part as we done above
                                                                                  and for second part it come after  decimal .For it just subtract
                                                                                  1 from remainder  here it is 4 so 4-1=3 and for second place
                                                                                   subtract 3 from 9 ,9-3=6  ie our answer now become . 36
                                                                                   This .36 repeat again  and again  ie .36363636......

                                                                           7             5              3             5
                                                                         6-6         14-8          13-5

                                                                           0            6               8              5

                                                                                0685.363636...... it is our required answer

                                                                           See one more example 7895
                                                                              Difference of odd position and even position

                                                                               7+9-8-5  =16 -13 =3 If we add 3 to 7895 it become
                                                                               fully divisible by 11 but our number is increased by 3
                                                                               so  we can not add 3  Here we adopt  another method
                                                                               instead of adding we subtract 3 from 11  and proceed as
                                                                                above  here our remainder is 11-3 =8
                                                                               So we subtract 8 from 7895
                                                                                7895     =    7887    +    8
                                                                                  11                11          11

                                                                                                       717     +   8
                                                                                                                      11   

                                                                                                            Here our remainder is 8 subtract 1 from it
                                                                                8-1=7  and this 7 is subtract from 9  9-7=2 so after decimal72 ....
                                                                                    ie  .727272...

                                                                            Required answer is 717.7272727....

                                                                        Be careful  in addition  and subtraction of remainders