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REMAINDER TRICKS


REMAINDER TRICKS



 Before going into the remainder tricks, I would like to tell u about Euler number.
What is Euler number and how to find out the Euler number of any other number?
          Answer of above question is very simple..
          
             Euler number is total number of numbers of co-prime that exist less then that number.
              For example    6 
                                      So 1 and 5 are the only co-prime which exists less then 6.
                                     So Euler number for 6=2.

          How to find Euler number of big numbers?

            First find out the total number of factors of that number and apply the following formula.
           Suppose we have a number p which has 3 factors q,r,s.
            Now to find the Euler number of p we will apply this formula :
                               
                                            Euler number= p * ((q-1)/q) * ((r-1)/r) * ((s-1)/s).

  For example Euler number of 21:
                           Factors of 21=3*7
                          So Euler number= 21 * ((3-1)/3) * ((7-1)/7)
                                                   =21* (2/3) * (6/7)
                                                   =2*6=12.

   Now we can easily calculate the Euler Number of any number.
             To apply these Euler number concept in finding remainder we will take a question i.e
                      *     Find out the remainder in 2^50/9 ?
                       Sol :      First calculate Euler number of 9 .
                                           Factor of 9=3*3
                                           Only one number i.e 3
                                           Now apply the formula i.e 9*((3-1)/3)
                                                                               =9*(2/3)
                                                                               =6.
                       So Euler number of 9 is 6.

                     Given problem was 2^50/9.
                     Now divide the power portion i.e 50 by Euler number i.e 6 = 50/6
                                                                                                                =remainder will be 2
                      Now replace the power by remainder
                                           
                                                  2^50/9
                                                =2^2/9
                                                =4/9
                                                = Remainder of it will be 4 answer
                                    



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