MEMO 2014 Problem 7

A finite set of positive integers $A$ is called meanly if for each of its nonempy subsets the arithmetic mean of its elements is also a positive integer. In other words, $A$ is meanly if $\frac{1}{k}(a_1 + \dots + a_k)$ is an integer whenever $k \ge 1$ and $a_1, \dots, a_k \in A$ are distinct.

Given a positive integer $n$ , determine the least possible sum of the elements of a meanly $n$ -element set.