If 2323 is divided by 53,then find the remainder.

hi,

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}}    

I have only one problem in the above so brilliantly done work out, whether the expansion (530 - 1) ¹¹ would end with +1 or a -1? If it ends with a +1, then the remainder is 23, but if it's -1, the remainder MUST be 53 - 23 = 30. Isn't it?