6. Determine all natural numbers $n$ for which the fraction $\frac{n+10}{n+2}$ is a natural number.
6. It holds that $\frac{n+10}{n+2}=\frac{n+2}{n+2}+\frac{8}{n+2}=1+\frac{8}{n+2}$

For the given fraction to be a natural number, the fraction $\frac{8}{n+2}$ must also be a natural number.

1 POINT The fraction $\frac{8}{n+2}$ will be a natural number if the denominator contains divisors of the number 8.

This means, $n+2$ can have the values $1, 2, 4$, and 8.

2 POINTS

$n+2=1$ has no solution in the set $\mathbf{N}$.

1 POINT

$n+2=2$ has no solution in the set $\mathbf{N}$.

1 POINT

$n+2=4$ gives $n=2$.

1 POINT

$n+2=8$ gives $n=6$.

1 POINT

Therefore, $n \in\{2,6\}$.