Let [itex]n[/itex] be a positive integer. Define a sequence by setting [itex]a_1 = n[/itex] and, for each [itex]k > 1[/itex], letting [itex]a_k[/itex] be the unique integer in the range [tex]0 \le a_k \le k-1[/itex] for which [itex]\displaystyle a_1 + a_2 + \cdots + a_k[/itex] is divisible by [itex]k[/itex]. For instance, when [itex]n = 9[/itex] the obtained sequence is [itex]\displaystyle 9, 1, 2, 0, 3, 3, 3, \ldots[/itex]. Prove that for any n the sequence [itex]\displaystyle a_1, a_2, a_3, \ldots[/itex] eventually becomes constant.(adsbygoogle = window.adsbygoogle || []).push({});

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# Difficult Proof

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