Question

$5^n+1$ is divisible by 6

• A. Even values of n
• B. Odd values of n
• C. All values of n
• D. Cannot say

Answer 1

The correct option is B Odd values of n

$5^n+1={(6-1)}^n+1$

${=}^n{C}_0{6}^n{-}^n{C}_1{6}^{n-1}+....+{(-1)}^{n-1}6{\times }^n{C}_{n-1}+{(-1)}^{n}6{\times }^n{C}_n+1$

$=6^n\left(k\right)+{(-1)}^n$ ${}^n{C}_n+1$ , where $k$ is an integer

$=6^n\left(k\right)+{(-1)}^n+1$

$=6^n\left(k\right)$, for odd values of $n$.

Hence, $B$ is correct.