If we have to find a set $S$ such that it has a given sum $n$ and it's LCM is maximum, then it is intuitive that the elements of the set should be coprime. However I can't find any formal proof. I have been able to prove that the numbers would be prime powers, but for coprimality I am completely stuck. I searched some, and faound that the LCM is known as Landau function of $n$, but I couldn't find any proof why the numbers should be coprime.
Edit: cardinality of set must be greater than 1, that's the only condition on the cardinality.