Chicken McNugget Theorem
The Chicken McNugget Theorem (or Postage Stamp Problem or Coin Problem) states that for any two relatively prime positive integers , the greatest integer that cannot be written in the form for nonnegative integers is .
A consequence of the theorem is that there are exactly positive integers which cannot be expressed in the form . The proof is based on the fact that in each pair of the form , exactly one element is expressible.
The story goes that the Chicken McNugget Theorem got its name because in McDonalds, people bought Chicken McNuggets in 9 and 20 piece packages. Somebody wondered what the largest amount you could never buy was, assuming that you did not eat or take away any McNuggets. They found the answer to be 151 McNuggets, thus creating the Chicken McNugget Theorem.
Definition. An integer will be called purchasable if there exist nonnegative integers such that .
We would like to prove that is the largest non-purchasable integer. We are required to show that (1) is non-purchasable, and (2) every is purchasable. Note that all purchasable integers are nonnegative, thus the set of non-purchasable integers is nonempty.
Lemma. Let be the set of solutions to . Then for any .
Proof: By Bezout's Lemma, there exist integers such that . Then . Hence is nonempty. It is easy to check that for all . We now prove that there are no others. Suppose and are solutions to . Then implies . Since and are coprime and divides , divides and . Similarly . Let be integers such that and . Then implies . We have the desired result.
Lemma. For any integer , there exists unique such that .
Proof: By the division algorithm, there exists such that .
Lemma. is purchasable if and only if .
Proof: If , then we may simply pick so is purchasable. If , then if and if , hence at least one coordinate of is negative for all . Thus is not purchasable.
Thus the set of non-purchasable integers is . We would like to find the maximum of this set. Since both are positive, the maximum is achieved when and so that .
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