What are the positive integers $n$ such that $n \mid n+7$?
Let $n$ be an integer such that $n \mid n+7$. Since $n \mid -n$, then $n \mid (n+7) + (-n)$, i.e., $n \mid 7$. The positive divisors of 7 are 1 and 7.

We then verify that $1 \mid 8$ and $7 \mid 14$. The solutions are therefore 1 and 7.