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Let $z$ be a complex number, and let $n$ be a positive integer such that $z^n = (z + 1)^n = 1$. Prove that $n$ is divisible by 6.
I feel like $|z|=1,$ but I don't know how to prove it. Even if I found that, how would it help? Solutions are greatly appreciated. Thanks in advance!
*I believe there's multiple ways to solve this question. And I would like to solve it using the one without much calculus. I only know up to intermediate algebra.