Question 189
If two real numbers $\alpha, \beta$ satisfy that $\lfloor k_1\alpha \rfloor \neq \lfloor k_2\beta \rfloor$ for all positive integers $k_1, k_2$, where $\lfloor x \rfloor$ denotes the maximal integer not exceeding $x$.
Prove that there exist two positive integers $m_1, m_2$ such that $\frac{m_1}{\alpha} + \frac{m_2}{\beta} = 1$.
Prove that there exist two positive integers $m_1, m_2$ such that $\frac{m_1}{\alpha} + \frac{m_2}{\beta} = 1$.