Question

What is the remainder when ${10}^{91}$ is divided by 13?

• A. 1
• B. 2
• C. none of these

Answer 1

The correct option is C none of these

$\frac{{10}^1}{13}{\mid }_R=10$
$\frac{{10}^2}{13}{\mid }_R=9$
$\frac{{10}^3}{13}{\mid }_R=(-1)$

Thus,
$\frac{{10}^6}{13}{\mid }_R=1$
Since, $\frac{{10}^6}{13}{\mid }_R=1=\frac{{10}^3\times {10}^3}{13}{\mid }_R=(-1)\times (-1)=1$

Hence we get ${10}^{90}{\mid }_R=1$, we are left with ${10}^1$ which will give a remainder of 10.
Hence the remainder when ${10}^{91}$ is divided by13 will be 10.