The remainder left out when 82n – (62)2n+1 is divided by 9 is

82n−(62)2n+1=(1+63)n−(63−1)2n+1 =(1+63)n+(1−63)2n+1 =(1+nc163+nc2(63)2+…+(63)n)+(1−(2n+1)c163+(2n+1)c2(63)2+….+(−1)(63)(2n+1)) =2+63(nc1+nc¯2(63)+…+(63)n−1−(2n+1)c1+(2n+1)c2(63)+….−(63)(2n))∴Remainder is 2.