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

## Comments

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

As an exercise, try to prove the theorem. – Mursalin Habib · 2 years ago

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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. – Calvin Lin Staff · 2 years ago

Considering thatLog in to reply

Frobenius numbers? – Agnishom Chattopadhyay · 2 years ago

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– Brian Kal · 2 years ago

Precisely. The chicken mcnugget theorem is a popularized version of frobenius though.Log in to reply

I feel sorry for McD... They will never be able to sell 151 nuggets by and large... – Nishant Prabhu · 2 years ago

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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) – Madhav Sainanee · 2 years ago

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– Indrajeet Dawle · 2 years ago

the two numbers has to be relatively prime to each other.Log in to reply

– Madhav Sainanee · 2 years ago

Ah..Thanks, my bad.Log in to reply

That's amazing. – Luke Zhuang · 2 years ago

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Applied math! – Bill Bell · 1 year, 3 months ago

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This is awesome.. – Daryll RomuaLdez · 2 years ago

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Then they decided to add 4 and 6 number of nuggets in a packet. So, what do I do? (weep...) – Aloysius Ng · 2 years ago

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– Pranjal Jain · 1 year, 9 months ago

Hah! They still cannot sell 151 nuggets!Log in to reply