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# 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
2 years, 7 months 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. · 2 years, 7 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 · 2 years, 7 months ago

Frobenius numbers? · 2 years, 7 months ago

Precisely. The chicken mcnugget theorem is a popularized version of frobenius though. · 2 years, 7 months ago

I feel sorry for McD... They will never be able to sell 151 nuggets by and large... · 2 years, 7 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) · 2 years, 7 months ago

the two numbers has to be relatively prime to each other. · 2 years, 7 months ago

Ah..Thanks, my bad. · 2 years, 7 months ago

That's amazing. · 2 years, 7 months ago

Applied math! · 1 year, 10 months ago

This is awesome.. · 2 years, 7 months ago

Then they decided to add 4 and 6 number of nuggets in a packet. So, what do I do? (weep...) · 2 years, 7 months ago