However, we defined to be a positive integer, and all positive integers are greater than or equal to. We can say that for some positive integer, and substitute to get. This can be rearranged into, which implies that is a multiple of (since ). If this is purchasable, we can say for some nonnegative integers. Proof: Any number that is less than and congruent to it can be represented in the form, where is a positive integer. Lemma: For any nonnegative integer, is the least purchasable number. Each of these permuted residues is purchasable (using the definition from Proof 1), because, in the form, is and is the original residue. Since, by the cancellation rule, that reduces to, which is a contradiction.'īecause and are coprime, we know that multiplying the residues of by simply permutes these residues. In other words,Ĭlearly none of the for are divisible by, so it suffices to show that all of the elements in are distinct. Then, we claim that the set, consisting of the product of the elements of with, taken modulo, is simply a permutation of. We start with this statement taken from Proof 2 of Fermat's Little Theorem: Since both are positive, the maximum is achieved when and so that. We would like to find the maximum of this set.

Thus the set of non-purchasable integers is. If, then if and if, hence at least one coordinate of is negative for all. Proof: If, then we may simply pick so is purchasable. Proof: By the division algorithm, there exists one and only one such that. For any integer, there exists unique such that. Since and are coprime and divides, divides and. Proof: By Bezout's Lemma, there exist integers such that. Note that all purchasable integers are nonnegative, thus the set of non-purchasable integers is nonempty. We would like to prove that is the largest non-purchasable integer. An integer will be called purchasable if there exist nonnegative integers such that. The Chicken McNugget Theorem has also been called the Frobenius Coin Problem or the Frobenius Problem, after German mathematician Ferdinand Frobenius inquired about the largest amount of currency that could not have been made with certain types of coins.ĭefinition. Math enthusiasts were curious to find the largest number of nuggets that could not have been bought with these packs, thus creating the Chicken McNugget Theorem (the answer worked out to be 151 nuggets). Originally, McDonald's sold its nuggets in packs of 9 and 20. However, the most popular by far remains that of the Chicken McNugget. There are many stories surrounding the origin of the Chicken McNugget theorem.

Chicken math definition