Question 66
a. For which positive numbers $n$ does there exist a sequence $x_{1}, x_{2}, \ldots, x_{n}$, which contains each of the numbers $1,2, \ldots, n$ exactly once and for which $x_{1}+x_{2}+\cdots+x_{k}$ is divisible by $k$ for each $k=1,2, \ldots, n$?
b. Does there exist an infinite sequence $x_{1}, x_{2}, x_{3}, \ldots$, which contains every positive integer exactly once and such that $x_{1}+x_{2}+\cdots+x_{k}$ is divisible by $k$ for every positive integer $k$?
b. Does there exist an infinite sequence $x_{1}, x_{2}, x_{3}, \ldots$, which contains every positive integer exactly once and such that $x_{1}+x_{2}+\cdots+x_{k}$ is divisible by $k$ for every positive integer $k$?