Function as Product of Functions

Suppose for all positive integers $a$ and $x$, let there be a function $f: \mathbb{Z^+} \to \mathbb{Z^+}$ such that $$f(ax)=f(x)⋅f(2x)⋅f(3x) \,...\, f((a-1)x)$$ find the least non-negative residue of $$f(100!) \bmod103$$