# Chicken McNugget Theorem

The Chicken Nugget Theorem, AKA the Postage Stamp Problem, states that for any two relatively prime positive integers $m,n$, the greatest integer that cannot be written in the form $am + bn$ for nonnegative integers $a, b$ is $\ mn-m-n$. (AoPS definition).

So, let's say you worked at a shop that sold cookies in packets of $13$ and $9$. What would be the largest number of cookies that you couldn't buy?

$m = 13$

$n = 9$

$(13 \times 9) - 13 - 9$

$=117 - 13 - 9$

Ans: $95$ cookies.

Origin:

A long time ago, in a galaxy far, far away...

McDonalds had sold Chicken McNuggets in packets of $9$ and $20$. Some wise person wondered what was the largest number of chicken nuggets that one couldn't buy. Later, the answer was found to be $151$ McNuggets. And thus, the Chicken McNugget Theorem had been formed.

~Brian

Note by Brian Kal
5 years ago

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Interesting. I didn't know that McDonalds had branches in other galaxies.

As an exercise, try to prove the theorem.

- 5 years ago

Considering that Dominos plans to open branch on the moon, I would not be surprised if McDonalds has branches on the Death Star. Those stormtroopers must have something to eat.

Staff - 5 years ago

Frobenius numbers?

Staff - 5 years ago

Precisely. The chicken mcnugget theorem is a popularized version of frobenius though.

- 5 years ago

That's amazing.

- 5 years ago

Hypothetically, if they sold two packs of 2 and 2 (sounds odd, but yeah) then according to the theorem you can't buy 0 nuggets. But the actual answer is 1(obviously)

- 5 years ago

the two numbers has to be relatively prime to each other.

- 5 years ago

- 5 years ago

I feel sorry for McD... They will never be able to sell 151 nuggets by and large...

- 5 years ago

Then they decided to add 4 and 6 number of nuggets in a packet. So, what do I do? (weep...)

- 5 years ago

Hah! They still cannot sell 151 nuggets!

- 4 years, 9 months ago

This is awesome..

- 5 years ago

Applied math!

- 4 years, 2 months ago