Question

The remainder when 23^23 is divided by 53 is:

Answer 1

23^23 = 23.(23)^22

           = 23 . ( 529)^11

            = 23. ( 530 -1 ) ^11

   ( 530 -1 ) ^11   =   nC0 530^11  +   nC 1 530^10   +........................................ +   nC n (1)^11

       23^23  =  23 . {nC0 530^11  +   nC 1 530^10   +........................................ +   nC n (1)^11 }

            first 10 terms will be divisible by 53 . remainder will come from the last term

          last term =   23

emplies that remainder is 23