Question

For all natural numbers n, ${10}^{2n-1}+1$ is divisible by

• A. 13
• B. 11
• C. 9
• D. none of these

Answer 1

The correct option is B

11

Substituting n=1, we have
${10}^1+1=11LetP\left(n\right):{10}^{2n-1}+1\text{is divisible by 11}$

P(1) is true.
Assume P(k) is true
$\Rightarrow {10}^{2k-1}+1=11mNow,P(k+1):{10}^{2k+1}+1={100.10}^{2k-1}+1={99.10}^{2k-1}+{10}^{2k-1}+111.({9.10}^{2k-1}+m)\to \text{divisible by 11}$

P(k+1) is also true.
Hence, P(n) is true.