Question

What will be the remainder when ${53}^{82}+53$ is divided by 54?

Answer 1

We have,

${53}^{82}+53$

$\Rightarrow 53({53}^{81}+1)$

$\Rightarrow 53({53}^{81}+1^{81})$

$\Rightarrow 53(53+1)({53}^{80}-{53}^{79}+{53}^{78}-{53}^{77}+.......+1)$

$\Rightarrow 53\times 54\times ({53}^{80}-{53}^{79}+{53}^{78}-{53}^{77}+.......+1)$

Then, the number is divisible by $54$. So the reminder is $0$.