MathNet 2005 — Part 02 Question 83

Let $a$, $b$ and $c$ be positive integers such that $ab$ divides $c\left(c^{2}-c+1\right)$ and $a+b$ is divisible by $c^{2}+1$. Prove that the sets $\{a, b\}$ and $\left\{c, c^{2}-c+1\right\}$ coincide.