MathNet 2012 Question 76

Find all integers $k \ge 3$, with the following properties: There exist integers $m$ and $n$ satisfying $(m, k) = (n, k) = 1$ and $k \mid (m-1)(n-1)$ with $1 \lt m \lt k$, $1 \lt n \lt k$ and $m+n \gt k$.