Maybe I have solved it, but I need only confirmation. Above expression is k/(n+k+2) . We need to find n (the smallest one) such as those expressions not to be further divided. And if you put n=k-7, this doesn’t work since for 9/(4+11)=9/15 what is divided by 3. Similar situation we have for n=k-6, n=k-5, n=k-4, n=k-2. Only it works for n=k-3 and we are looking for smallest "n" and we end up on this n=k-3, it doesn't matter what is going with n=k-1. Moreover this n=k-3 valid for all k belong to N, not only for restricted k €(7,8,9,...31) as it is given by this assignment above.
May you help me to prove it. I think it is induction method which should be useful to be applied, but I am not so familiar how to proceed it.