Fix an integer $n \ge 2$. A finite set $A \subset \mathbb{N} $ is given.
Define $ s(X) = \sum_X x $, where $ X $ is a finite set. We know that $n \mid s(A)$.
We can do just one move: if there is a subset $S \subseteq A$ such that $n \mid s(S)$, then we can take out an arbitrary element of $S$. To reduce $A$ means to make a sequence of moves such that no more moves are possible.
We call reduction degree of $A$ with respect to $n$ the minimum $d \in \mathbb{N}$ such that exists a sequence of $d$ moves which reduces $A$.
Is there a human way to explicitly find the degree of reduction of a set? This is deeply linked with a problem of number theory I'm approaching with a friend of mine. Clearly, partial solutions of subcases would be appreciated.