Diophantine Equation and Euclid's algorithm

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In summary, the equation 13x + 4y = 100 has a solution of x = 4 and y = 12, which results in x + y = 16. This can be found using Euclid's algorithm, but there is an easier way to solve the equation by noticing that x must be an even number and choosing x = 4 to guarantee that y is also an integer.
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Spruance
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Given that x and y are positive integers such that:

13x + 4y = 100

Then, what is x + y like?

Personally, I found the answer using Euclid's algorithm.


d = gcd(13,4)=1

13 * u + 4 * v = 1

(u, v) = (1,-3)

(x,y) = (100, -300)

13 * (x - 100) = 4 * (y + 300)

(Gauss)

x = 4 * k & y = 13 * k - 1

we set k = 1;

ergo is x = 4 and y = 12

x + y = 4 + 12 = 16

However, my teacher told me that it exist a much easier way to solve the equation. Anyone that know his solution to the problem?

Thanks in advance
 
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  • #2
Spruance said:
Given that x and y are positive integers such that:

13x + 4y = 100

Then, what is x + y like?

Personally, I found the answer using Euclid's algorithm.


d = gcd(13,4)=1

13 * u + 4 * v = 1

(u, v) = (1,-3)

(x,y) = (100, -300)

13 * (x - 100) = 4 * (y + 300)

(Gauss)

x = 4 * k & y = 13 * k - 1

we set k = 1;

ergo is x = 4 and y = 12

x + y = 4 + 12 = 16

However, my teacher told me that it exist a much easier way to solve the equation. Anyone that know his solution to the problem?

Thanks in advance


13x +4y = 100 and x and y are integers

first off if this is to be true then x HAS to be a positive even number.
also notice that 13*8 is 80+24=104 therefore x can only be 2 4 or 6.

Next, 13 and 4 have no common factors therefore we must choose that x is equal to 4 for it to be guaranteed that y is also an integer. So if x = 4, then y is 100/4 - 13 = 12.

Hope this helps!
 

1. What is a Diophantine equation?

A Diophantine equation is a polynomial equation in two or more unknowns with integer coefficients. It is named after the ancient Greek mathematician Diophantus, who studied and wrote about such equations.

2. How is Euclid's algorithm used to solve Diophantine equations?

Euclid's algorithm is a method for finding the greatest common divisor (GCD) of two integers. It can be used to solve Diophantine equations by finding the GCD of the coefficients of the variables, and then using this GCD to find a solution to the equation.

3. Can all Diophantine equations be solved using Euclid's algorithm?

No, not all Diophantine equations can be solved using Euclid's algorithm. This method only works for equations with two or more unknowns and integer coefficients. Other types of equations may require different methods of solution.

4. What is the significance of solving Diophantine equations?

Solving Diophantine equations has applications in many areas of mathematics, including number theory, algebra, and geometry. These equations also have real-world applications, such as in cryptography and coding theory.

5. Are there any limitations to using Euclid's algorithm for solving Diophantine equations?

One limitation of Euclid's algorithm is that it only provides one solution to a Diophantine equation, even if there are multiple solutions. Also, the algorithm becomes more complex and time-consuming as the number of variables in the equation increases.

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