That's actually really cool!

$\lfloor \alpha \rfloor,\lfloor 2\alpha \rfloor, \lfloor 3\alpha \rfloor, \ldots \quad \text{ and } \quad \lfloor \beta \rfloor, \lfloor 2\beta \rfloor,\lfloor 3\beta \rfloor, \ldots$

Let $\alpha$ and $\beta$ be positive irrational numbers such that the sequences above contain every positive integer exactly once. i.e. If you pick a positive integer, it will appear in only one of the above sequences. Which of the following statements are true?

Notation: $\lfloor \cdot \rfloor$ denotes the floor function.