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Puzzle 84: Chicken Nuggets

Picture
You will need to show detail proof on how you arrived at the largest number you cannot buy.

Extension: Using the same proof and logic, what if you add boxes of other quantities into the mix?  Say 15 or 30?  What is now the largest number of nuggets you cannot buy?  Explain and proof.
Source: http://www.playwithyourmath.com/