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sum of n different integers divisible by n

Source: Nordic Mathematical Contest 1998 #3

October 3, 2017
number theoryInteger sequenceDivisibility

Problem Statement

(a) For which positive numbers nn does there exist a sequence x1,x2,...,xnx_1, x_2, ..., x_n, which contains each of the numbers 1,2,...,n1, 2, ..., n exactly once and for which x1+x2+...+xkx_1 + x_2 +... + x_k is divisible by kk for each k=1,2,....,nk = 1, 2,...., n? (b) Does there exist an infinite sequence x1,x2,x3,...,x_1, x_2, x_3, ..., which contains every positive integer exactly once and such that x1+x2+...+xkx_1 + x_2 +... + x_k is divisible by kk for every positive integer kk?
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