MathNet 2020 — Math Olympiad Vault — Part 02 Question 55

On a train returning from EGMOnd an Zee, Clara and Edwige play a game. Initially $n=1 \times 2 \times \cdots \times 20$ is written on paper. They alternate, Clara first, replacing the integer $n$ by $k n / 10$, where $k$ is an integer from 1 through 9. The first player to write a noninteger loses.
Both play optimally. Who wins?