Question 173
Let $n$ be a positive integer. Define a sequence by setting $a_1 = n$ and, for each $k \gt 1$, letting $a_k$ be the unique integer in the range $0 \le a_k \le k - 1$ for which $a_1 + a_2 + \cdots + a_k$ is divisible by $k$. For instance, when $n = 9$ the obtained sequence is $9, 1, 2, 0, 3, 3, 3, \dots$. Prove that for any $n$ the sequence $a_1, a_2, a_3, \dots$ eventually becomes constant.