Question

What is the remainder when $3^{57}$ $+27$ is divided by $28$?

Answer 1

$3^{57}=(3^3{)}^{39}$
$\therefore 3^{57}+27=(27{)}^{19}+27$
$=(28-1{)}^{19}+27$
$28M+(-1{)}^{19}+27$ [Expand by binomial theorem]
$=28M-1+27$
$=28M+26$
When $28M+26$ divided by $28$ the remainder is $26$ Hence the required remainder is $26$.