This game is equivalent to a "normal game of Nim" (https://en.wikipedia.org/wiki/Nim) starting with n heaps each having an initial size equal to the sum of the exponents in the factorization of m. It just so happens that a very terse simplified solution exists for n heaps of the same initial size, but that is not trivial to prove. I doubt the authors realize the complexity of this puzzle, when done right.