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({});

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Difficult Proof

**Physics Forums | Science Articles, Homework Help, Discussion**