Question

Find the remainder when $8^{2n}-(62{)}^{2n+1}$ is divided by $9$

Answer 1

$8^{2n}=(8^2{)}^n={64}^n=(1+63{)}^n$

$=63k+1$ for $KϵN$

$(62{)}^{2n+1}=(63-1{)}^{2n+1}=63m-1$ for $mϵN$

$8^{2n}-\left(62\right)2n+1$

$(63K+1)-(63m-1)$

$63(K-m)+2$

$q\times 7(k-m)+2$

$qp+2,pϵN$

$\therefore$ required remainder is 2