MathNet 2026 — Part 08 Question 87

Let $k$ and $n$ be positive integers such that $k \lt 2^{n+1} - 1$. Prove that there is a sum of exactly $n$ powers of $2$ that is divisible by $k$. (Example: if $k = 9$ and $n = 4$, then $2 + 4 + 16 + 32$ is divisible by $9$.)