Partitioning a Set so that the Sum is the Same Let $M = \{1, 2, \dots , n\}$. What would be necessary and sufficient condition(s) for the number $m$, so that $M$ can be expressed as the disjoint union of $m$ subsets $A_i$, $(i = 1, 2, \dots, m$), such that
(i) each $A_i$ contains the same number of elements, and
(ii) the sum of all elements of $A_i$ is the same for $i = 1, 2, \ldots, m.$
Obviously, $m\mid n$. Other than that, I don't know where to begin!
 A: Since by (ii), the sum of the elements of $A_i$ is the same for every $i$, let's call this sum $S$.


*

*Now we have $m$ sets each of which has elements that add up to $S$.  That means that all together they add up to...?

*But the sets are all disjoint, and together they contain all the elements of $M$, which is $\{1,2,\ldots,n\}$, so the sum of all of them together is the same as the sum of the elements of $M$, which is...?

*But both (1) and (2) are two different ways of adding up the same thing, namely all the elements of the $A_i$, so those two sums must be equal, which gives you an equation, which is ...?

*Each of the $A_i$ has the same number of elements.  Say this number is $e$.  Then since all together the $A_i$ have the sane number of elements as $M$ does, we can relate $e$ and $n$ and $m$ with the equation...?
I hope this gives you some ideas about where to begin.
An important principle at work here, which your class may not have pointed out, is that often there is some important piece of information, such as the number of elements in an $A_i$, or the sum of those elements, which doesn't have a name. If you can identify these important components, and name them, you are making progress.
