Answer
$7^n-1$ is divisible by 6
Work Step by Step
We can find an expression for $7^n-1$:
$7^n-1$
$= (6+1)^n-1$
$= (a_1~6^n+a_2~6^{n-1}+a_3~6^{n-2}+...+a_n~6+1)-1$
$= a_1~6^n+a_2~6^{n-1}+a_3~6^{n-2}+...+a_n~6$
$= (6)(a_1~6^{n-1}+a_2~6^{n-2}+a_3~6^{n-3}+...+a_n)$
Since this number is a multiple of 6, it must be divisible by 6.
Note that the coefficients $a_i$ are positive integers.
