Solution:

We describe a winning strategy for Anne. Her first move is
$$10^{2007} \rightarrow 2^{2007}, 5^{2007}$$
We want to show that Anne can act in such a way that the numbers on the blackboard after each of her moves are of the form
$$2^{\alpha_{1}}, \ldots, 2^{\alpha_{k}}, 5^{\alpha_{1}}, \ldots, 5^{\alpha_{k}}$$
This is the case after Anne's first move. If Berit for example replaces $2^{\alpha_{j}}$ by $2^{\beta_{1}}$ and $2^{\beta_{2}}$, then Anne would replace $5^{\alpha_{j}}$ by $5^{\alpha_{1}}$ and $5^{\alpha_{2}}$. If Berit for example erases $5^{\alpha_{j}}$ or two $5^{\alpha_{j}}$'s (which means that there is an $\alpha_{i} = \alpha_{j}$), then Anne would erase $2^{\alpha_{j}}$ or two $2^{\alpha_{j}}$'s. Thus for each move Berit makes, Anne can answer with a 'symmetric' move. Since the game is finite, Berit must be the first player failing to make a move. Thus Anne has a winning strategy.