Question

If the remainder when $x$ is divided by 4 is $3,$ then the remainder when $(2020+ x )^{2022}$ is divided by 8 is __________.

Answer 1

$x=4 k+3$
$\therefore (2020+ x )^{2022}=(2020+4 k +3)^{2022}$
$=(4(505+ k )+3)^{2022}$
$=(4 \lambda+3)^{2022}=\left(16 \lambda^{2}+24 \lambda+9\right)^{1011}$
$=\left(8\left(2 \lambda^{2}+3 \lambda+1\right)+1\right)^{1011}$
$=(8 p +1)^{1011}$
$\therefore $ Remainder when divided by $8=1$