What I learned from this reading is if two integers x and y and both even or both odd they are of the same parity, and if one is odd while the other is even they are of different parity. I always thought this was something but I didn't know the specific name that it had assigned to this property. Another tip I thought was important was when you are evaluating a proof you need to solve it for the general case, and you can't just assign a specific number that works for the proof, or you haven't solved it for all cases.
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