Question

The set of integers $n$ for which $2^n+1$ is divisible by $7$ is

• A. $\{52n|n\in N\}$
• B. $\{77n|n\in N\}$
• C. $\{2^n-1|n\in N\}$
• D. none of these

Answer 1

Answer: (D)

none of these

$2^n+1$
$=(2^3{)}^{\frac{n}{3}}+1$
$=(7+1{)}^{\frac{n}{3}}+1$
$=7m+(1{)}^{\frac{n}{3}}+1$
Hence n has to be a multiple of 3.
Therefore
$2^n+1$ becomes
$7m+2$
By dividing it by 7, we get a remainder as 2.
Hence the answer is none of these.