Question

When $5^{99}$ is divided by $13$, the remainder is

• A. $8$
• B. $9$
• C. $10$
• D. none of the above

Answer 1

Answer: (A)

$8$

We have $5^{99}=5^3\times 5^{96}=\left(125\right){\left(625\right)}^{24}$

$=[13\times 9+8]{(1+48\times 13)}^{24}=[13\times 9+8][1{+}^{24}{C}_1\times (48\times 13){+}^{24}{C}_2\times {(48\times 13)}^2+..+{(48\times 13)}^{24}]$

$=8+$ term containing powers of $13$

Hence remainder $=8$.