For $n > 5$ and $1 < k < n+1$ prove that $n! + k$ has a prime divisor that is greater than $n$.
This question appears as an exercise in the book "Not always buried deep" by Paul Pollack. There is a reference, but the paper is in German! It seems to be enough to prove the existence of a prime divisor of $n!+k$ that is greater than $k$ for if all prime divisors of $n!+k$ were less than $n+1$, then they would divide n! and hence would have to divide $k$. The case $k = 1$ is easy to handle, but am stuck for $k > 1$.