Question

The statement $P\left(n\right)=5^{2n}+1$ is divisible by $13$ is true, then $5^{99}$ leaves a remainder, when divided by $13$, is

• A. $7$
• B. $6$
• C. $8$
• D. None of these

Answer 1

Answer: (C)

$8$

Given $5^{2n}+1$ is divisible by 13.

Let $5^{99}=x$.

$x=5^{99}=5^{2\left(49\right)}.5+5-5=5(5^{2\left(49\right)}+1)-5=5\left(13k\right)-5$

remainder of $x=13-5$ since the first term is divisible by $13$

$\therefore remainder=8$