How should I solve this Diophantine equation word problem?

In summary, the conversation discusses finding solutions in non-negative integers for the Diophantine equation 3x+2y+0.5z=100, where x represents the number of men, y represents the number of women, and z represents the number of children. By substituting z=100-x-y into the equation and using the Euclidean Algorithm, it is determined that the equation can be solved. The conversation then mentions using modulo arithmetic to find solutions and notes that this problem is from the section of Diophantine equations.
  • #1
Math100
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Homework Statement
Alcuin of York, 775. One hundred bushels of grain are distributed among 100 persons in such a way that each man receives 3 bushels, each woman 2 bushels, and each child 1/2 bushel. How many men, women, and children are there?
Relevant Equations
None.
Proof: Let x be the number of men, y be the number of women
and z be the number of children.
We need to find the solutions in the non-negative integers
for the Diophantine equation 3x+2y+0.5z=100 such that
x+y+z=100.

From x+y+z=100, we have that z=100-x-y.
Substituting z=100-x-y into the Diophantine equation
3x+2y+0.5z=100 and multiplying it by 2 produces:
5x+3y=100.

Applying the Euclidean Algorithm produces:
5=1(3)+2
3=1(2)+1
2=2(1)+0.

Now we have that gcd(5, 3)=1.
Note that 1##\mid##100.
Since 1##\mid##100, it follows that the Diophantine equation
5x+3y=100 can be solved.

Then we have 1=3-1(2)
=3-1(5-1(3))
=2(3)-1(5)

And now I'm stuck on this problem. I know I need to find the x0 and y0 in order to seek the general solution of the Diophantine equation. How should I go from here?
 
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  • #2
From ##3x +2y = 100## you could do some modulo arithmetic.

PS It shold be ##5x +3y = 100##, of course!

There are lots of solutions.
 
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  • #3
The modern version of this problem has bitcoins instead of bushels.
 
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  • #4
But this problem is from the section of Diophantine equations. Would it still be okay to do some modulo arithmetic without doing the back substitution for the Diophantine equation 5x+3y=100?
 
  • #5
Math100 said:
But this problem is from the section of Diophantine equations. Would it still be okay to do some modulo arithmetic without doing the back substitution for the Diophantine equation 5x+3y=100?
Modulo arithmetic is as Diophantine as it gets!
 

FAQ: How should I solve this Diophantine equation word problem?

1. How do I approach solving a Diophantine equation word problem?

There are a few steps you can follow to solve a Diophantine equation word problem. First, identify the unknown variables and assign them letters. Then, translate the word problem into an equation or system of equations. Next, use algebraic techniques such as substitution or elimination to solve for the variables. Finally, check your solution to make sure it satisfies the original problem.

2. What are some common techniques for solving Diophantine equations?

Some common techniques for solving Diophantine equations include factoring, completing the square, and the quadratic formula. Additionally, using number properties such as divisibility rules and prime factorization can also be helpful in solving these types of equations.

3. Can I use a calculator to solve Diophantine equations?

While a calculator can be useful for checking your work, it is generally not recommended to use one to solve Diophantine equations. These types of equations involve using algebraic techniques and critical thinking skills, which cannot be replaced by a calculator.

4. What are some real-life applications of Diophantine equations?

Diophantine equations have many real-life applications, particularly in fields such as cryptography, engineering, and computer science. They can be used to solve problems involving distance, time, and other variables, as well as in optimizing solutions for various systems.

5. Are there any tips for solving Diophantine equations more efficiently?

One tip for solving Diophantine equations more efficiently is to practice, as these types of problems often require creative thinking and practice can help improve your problem-solving skills. Additionally, breaking down the problem into smaller, more manageable steps can also make the process more efficient.

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