No point in dealing with all the prime factors at once. Define the p-adic map: so essentially our problem reduces to
In less formal terms, for every element subset of we take its smallest element, and we add it to our total if is odd and subtract from our total if is even, and we want to prove after doing this for all subsets we end up with the biggest element of

WLOG let Now consider how many times is added in the right hand side. If we can show it is added a net total of zero times for but just once when our life is complete.

Consider the elements In the first summation, is obviously added time. In the second summation, it's subtracted times, once every time is compared to one of Similarly, the number of times it's added is times, which is equal to iff and zero otherwise. And we're done!

Don't even have time to attempt the harder and more interesting problems cause of school work. >.<

One thing I realized is that the Combo 3 problems are of good quality so I should probably run through the harder ones during winter break.

Ok back to the torture chamber.