Product of Coprime Factors

$337500 = 2^2 \cdot 3^3 \cdot 5^5$

So, since the $a$ and $b$ must be coprime, then they can't have any of the same prime factors.

This gives us $\boxed{4}$ unordered ways to choose $(a, b)$ :

• $(1,337500)$
• $(2^2, 3^3 5^5)$
• $(3^3, 2^2 5^5)$
• $(5^5, 2^2 3^3)$