Solving Diophantine Equations: Find Largest Number of Chicken Nuggets

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In summary, the conversation was about a math problem involving buying chicken nuggets in sets of 6, 9, and 20 and finding the largest number of nuggets that cannot be bought. The person telling the story struggled with the problem at first and used the guess and check method to find the answer. They later realized that the problem could be solved using modulo arithmetic, but could not remember the exact reasoning behind it. The conversation also mentioned a teacher who gave extra credit for attempting the problem.
  • #1
InsaneMind
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So the other day I was asked a question about a problem by a friend. Having not been active in a math class in a VERY VERY long time I struggled with it at first... I believe it was worded as such...

Nick goes to McDonald's to buy chicken nuggets. Nick can only buy chicken nuggets in sets of 6, 9 and 20. After a bit, nick tries to figure out what's the largest number of chicken nuggets nick "cannot" buy. Help nick find out his answer, you can use any method you like (including guess and check)


Now originally I recognized that the problem is...

6a + 9b + 20c = n

Which I remembered as a diophantine equation, however I couldn't remember the formula to solve the problem. So I began using the guess and check method and arrived at the appropriate answer...checked his teachers guide to verify and then showed him how I did my guess and check method. His teacher gave out points to students whether they attempted the problem or not as long as they provided substantial work into it and/or arrived at an answer. (Basically as long as he tried he would get the extra-credit...I wish I had that teacher back in the day).


My question is...what's the proper formula for finding the answer?


The answer was 43
 
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  • #2
I read about this problem some time back before. I don't think it has anything to do with Diophantine equation, it's more of modulo arithmetic. Except I can't recall the reasoning too.
 

1. How do you approach solving Diophantine equations?

To solve Diophantine equations, we typically use a combination of algebraic techniques and number theory concepts. This involves manipulating the given equation to isolate a single variable and then using properties of integers such as divisibility and prime factorization to find suitable solutions.

2. What is a Diophantine equation?

A Diophantine equation is a polynomial equation in two or more unknowns with integer coefficients. These equations are named after the ancient Greek mathematician Diophantus, who was known for his work in solving these types of equations.

3. How do you find the largest number of chicken nuggets that can be purchased?

The first step is to set up a Diophantine equation representing the total number of chicken nuggets that can be purchased. This involves using the number of nuggets in each package and the total amount of money available. Then, we can use techniques such as the Euclidean algorithm and modular arithmetic to find the largest possible solution.

4. Are there any real-world applications of Diophantine equations?

Yes, Diophantine equations have many real-world applications in fields such as cryptography, engineering, and computer science. They are also used to model and solve problems in number theory, algebra, and geometry.

5. Are there any special cases of Diophantine equations that are easier to solve?

Yes, there are certain types of Diophantine equations that have well-known solutions and can be solved using specific techniques. These include linear Diophantine equations, which involve only linear terms, and quadratic Diophantine equations, which involve a single quadratic term.

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