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Part of 2006 Korea National Olympiad

Problems(1)

Factoring Game

Source: 2006 Korea National Olympiad #2

3/18/2018
Alice and Bob are playing "factoring game." On the paper, 270000(=243354)270000(=2^43^35^4) is written and each person picks one number from the paper(call it NN) and erase NN and writes integer X,YX,Y such that N=XYN=XY and gcd(X,Y)≠1.\text{gcd}(X,Y)\ne1. Alice goes first and the person who can no longer make this factoring loses. If two people use optimal strategy, prove that Alice always win.
primesCombinatorial gamescombinatorics