$6^{ n } + 8^{ n }$ is divisible by 7 if:

$n$
is a prime number
$n$
is any real number
$n$
is an odd number
$n$
is an even number

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$6^n+8^n = (7-1)^n+(7+1)^n \equiv (-1)^n+1^n \pmod {7}$

$(-1)^n + 1^n = 0$ when $\boxed {n \space is \space odd }$ .