Question 489206: can the set {1,2,...,2010) be partitioned into classes A1 ,A2 ,..,An such each of the classes contains the same number of elements and the sum of elements in each classes are the same
Answer by chessace(471)   (Show Source):
You can put this solution on YOUR website! Yes, if and only if n is a divisor of 2010: n=1,2,3,5,67.
Because there are an even number in the entire set, any divisor n of 2010 will allow you to creat n subsets of equal size and with equal totals.
Use the "Gauss in grade school" trick to ensure that all subsets total a multiple of 2011:
A1 gets 1 and 2010, A2 gets 2 and 2009, etc., repeating A1 after An in this cycle.
n=1 is a trivial case, but meets the requirements.
Any n not a divisor will fail to have the same size sets.
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