MathNet 2010 — Part 04 Question 98

Let $k,l$ be distinct natural numbers, and let $a_1,\ldots,a_n$ be nonzero complex numbers, not necessarily distinct. If $\{a_1^k,\ldots,a_n^k\}=\{a_1^l,\ldots,a_n^l\}$, prove that every $a_i$ is a root of unity.