82n−622n+1=(a−1)2n−(63−1)2n+1 =(2nc092n−2nc192n−1+2nc292n−2..... −2nc2n−1(9)+2nc2n) −2n+1c0(63)2n+1−2n+1c1(63)2n +2n+1c2(63)2n−1........... +2n+1c1(63)−2n+1c0 =9m+1+1=9m+2 for some integer m.
Thus 82n−622n+1 is divided by 9, the remainder is 2.