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

$\begin{array}{rl} & 8^{2\mathrm{n}}-(62{)}^{2\mathrm{n}+1}=(1+63{)}^{\mathrm{n}}-(63-1{)}^{2\mathrm{n}+1}\\ & =(1+63{)}^{\mathrm{n}}+(1-63{)}^{2\mathrm{n}+1}\\ & =\left(1{+}^{\mathrm{n}}{\mathrm{c}}_{1}63{+}^{\mathrm{n}}{\mathrm{c}}_2\right(63{)}^2+\dots +\left(63{)}^{\mathrm{n}}\right)+\left(1{-}^{(2\mathrm{n}+1)}{\mathrm{c}}_{1}63{+}^{(2\mathrm{n}+1)}{\mathrm{c}}_2\right(63{)}^2+\dots .+(-1)\left(63{)}^{(2\mathrm{n}+1)}\right)\\ & =2+63{(}^{\mathrm{n}}{\mathrm{c}}_1{+}^{\mathrm{n}}{\overline{\mathrm{c}}}_2\left(63\right)+\dots +(63{)}^{\mathrm{n}-1}{-}^{(2\mathrm{n}+1)}{\mathrm{c}}_1{+}^{(2\mathrm{n}+1)}{\mathrm{c}}_2\left(63\right)+\dots .-\left(63{)}^{\left(2\mathrm{n}\right)}\right)\end{array}$

$\therefore$Remainder is 2.