The remainder when 82n−622n+1is divided by 9 is

Solution:

$8^{2 n} - 62^{2 n + 1}$

$\begin{array}{l} = (9 - 1)^{2 n} - (63 - 1)^{2 n + 1} \\ = (^{2 n} C_{0} 9^{2 n_{-} 2 n} C_{1} 9^{2 n - 1} +^{2 n} C_{2} 9^{2 n - 2} - \dots - \\ ^{2 n} C_{2 n - 1} (9) +^{2 n} C_{2 n}) - (^{2 n + 1} C_{0} (63^{2 n + 1}) -^{2 n + 1} C_{1} (63^{2 n}) \\ +^{2 n + 1} C_{2} (63^{2 n - 1}) - \dots - +^{2 n + 1} C_{1} (63) -^{2 n + 1} C_{0}) \end{array}$

$= 9 m + 1 + 1 = 9 m + 2$ for some integer $m$

Thus, when $3^{2 n} - 62^{2 n + 1}$

is divided by, $9$ the remainder is $2$ .

The remainder when $8^{2 n} - 62^{2 n + 1}$ is divided by $9$ is

Solution:

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