Explanation

(b)
$8^{2n} - (62)^{2n + 1} = (1 + 63)^{n} - (63 - 1)^{2n + 1}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = (1 + 63)^{n} + (1 - 63)^{2n + 1}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \left( 1 + \ ^{n}C_{1}63 + \ ^{n}C_{2}(63)^{2} + \ldots + (63)^{n} \right)$
$+ {(1 -}^{(2n + 1)}C_{1}63 +^{(2n + 1)}C_{2}(63)^{2} + \ldots + ( - 1)\left( {63)}^{(2n + 1)} \right)$
$= 2 + 63\lbrack\ ^{n}C_{1} + \ ^{n}C_{2}(63) + ... + (63)^{n - 1}{\ _{-}}^{(2n + 1)}C_{1}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +^{(2n + 1)}C_{2}(63) - ... - {(63)}^{(2n)}\rbrack$
$\therefore$ Remainder is 2.