MathDB
p-partitionable

Source: Ireland 1997

July 3, 2009
combinatorics unsolvedcombinatorics

Problem Statement

Let p p be an odd prime number and n n a natural number. Then n n is called p\minus{}partitionable if T\equal{}\{1,2,...,n \} can be partitioned into (disjoint) subsets T1,T2,...,Tp T_1,T_2,...,T_p with equal sums of elements. For example, 6 6 is 3 3-partitionable since we can take T_1\equal{}\{ 1,6 \}, T_2\equal{}\{ 2,5 \} and T_3\equal{}\{ 3,4 \}. (a) (a) Suppose that n n is p p-partitionable. Prove that p p divides n n or n\plus{}1. (b) (b) Suppose that n n is divisible by 2p 2p. Prove that n n is p p-partitionable.
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