Question 24
Let $n$ and $k$ be positive integers such that $n \lt k \lt 2n$. Suppose $a_1, a_2, \ldots, a_k$ are positive integers such that $2^{a_1} + 2^{a_2} + \cdots + 2^{a_k}$ is divisible by $2^n - 1$.
Show that at least three of $a_{1}, a_{2}, \ldots, a_{k}$ have the same remainder when divided by $n$.
Show that at least three of $a_{1}, a_{2}, \ldots, a_{k}$ have the same remainder when divided by $n$.