# 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
6 years, 8 months ago

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Sort by:

Applied math!

- 5 years, 11 months ago

- 2 months, 3 weeks ago

This is awesome..

- 6 years, 8 months ago

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

- 6 years, 8 months 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)

- 6 years, 8 months ago

- 6 years, 8 months ago

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

- 6 years, 8 months ago

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

- 6 years, 8 months ago

Hah! They still cannot sell 151 nuggets!

- 6 years, 5 months ago

That's amazing.

- 6 years, 8 months ago

Frobenius numbers?

- 6 years, 8 months ago

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

- 6 years, 8 months ago

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

As an exercise, try to prove the theorem.

- 6 years, 8 months 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 - 6 years, 8 months ago