Question 4
Let $w = \frac{\sqrt{3} + \mathrm{i}}{2}$ and $z = \frac{-1 + \mathrm{i}\sqrt{3}}{2},$ where $\mathrm{i} = \sqrt{-1}.$ Find the number of ordered pairs $(r, s)$ of positive integers not exceeding $100$ that satisfy the equation $\mathrm{i} \cdot w^r = z^s.$