Question

Find the remainder when $5^{99}$ is divided by $8$ is

Answer 1

$5^{99}=5(5^2{)}^{49}$
$=5(24+1{)}^{49}$
$=5{(}^{49}{C}_0{24}^{49}{+}^{49}{C}_1{24}^{48}+\cdots {+}^{49}{C}_{48}24+1)$

Hence remainder when $5^{99}$ is divided by $8$ is $5$