For a given positive integer $n$, how long can the process $x \mapsto n\text{ }(\text{mod } x)$ last before reaching $0$? We improve Erdős and Shallit's upper bound of $O(n^{\frac{1}{3}+\varepsilon})$ to $O(n^{\frac{1}{3}-\frac{2}{177}+\varepsilon})$ for any $\varepsilon > 0$.
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