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$2k$ divides $n(n+1)$ and $2k\leq n+1$

Source: Moldova TST 2001

August 7, 2023
number theory

Problem Statement

Let nn{} (n≥1)(n\geq 1) be an integer and a set A={1,2,…,n}A=\{1,2,\ldots,n\}. The set AA{} is k−partitionablek-partitionable if it can be partitioned in kk{} disjoint sets with the same sum of elements. Show that AA{} is k−partitionablek-partitionable if and only if 2k2k divides n(n+1)n(n+1) and 2k≤n+12k\leq n+1.
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