Problem of the Week #1

(Note: the math symbols in this post will come out more cleanly if you open up the single post by itself, rather than viewing it within the whole blog.)

Welcome to the Austin Math Circle problem of the week. Each week we'll put up a new problem here. (Mostly drawn from old contests, but maybe some original ones from time to time.) Put answers/solutions in the comments, and the next week we'll put up a solution, along with discussion of any particularly interesting approaches that people took.

The number 102007 is written on the blackboard. Anne and Berit play a two-player game in which players alternate performing one of the following operations:
  1. Replace any number x on the blackboard with two integers a, b > 1 such that ab = x.
  2. Erase one or both of any two equal numbers on the blackboard.
The person who first cannot perform any operations loses the game.

Who has the winning strategy if Anne moves first, and why?

[Source: Nordic Mathematical Competition 2007, Q3. But don't look! Solve it on your own.]

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