Solving Linear Diophantine Equation in Discrete Mathematics

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In Number Theory, we study Linear Diophantine equations—equations that seek whole number solutions. These typically take the form ax + by = c, where a, b, and c are known values, and x and y are the integers to be determined. Diophantine equations have applications in various fields, including number theory and cryptography. They are particularly useful for solving problems that require integer-only solutions.

In this chapter, we will explore the basics of solving these equations, analyze examples, and understand the conditions necessary for solutions.

Linear Diophantine Equation

A linear Diophantine equation takes the form ax + by = c, where a, b, and c are the given integers. The equation is called "Diophantine" because we are only interested in integer solutions for x and y. Not all Diophantine equations have solutions because of this integer constraint. Also these equations may have multiple answers.

For a solution to exist, there should be one key rule: the greatest common divisor (GCD) of a and b must divide c. If it does, there are infinitely many solutions; But if it does not hold, then the equation has no solution.

Example

Consider the equation 3x + 6y = 12. Here, the GCD of 3 and 6 is 3. This GCD is factor of 12. So we know that there are solutions. However, for 4x + 8y = 10, the GCD of 4 and 8 is 4.

Basic Steps for Solving Linear Diophantine Equations

To solve ax + by = c, we generally follow these steps:

• Check the GCD Condition − Ensure that the GCD of a and b is factor of divides c.
• Find a Particular Solution − Use the extended Euclidean algorithm to find one set of values for x and y.
• Generate the General Solution − Since there are infinitely many solutions, we express them in terms of an integer parameter k.

Let us see the steps in detail using examples.

Checking the GCD Condition

The first task we perform is checking if the equation has any solutions by finding the GCD of a and b. If the GCD divides c, then we can proceed with finding solutions.

Example

Take the equation 10x + 15y = 35. Here, the GCD of 10 and 15 is 5. This divides 35, so we know solutions exist.

Finding a Particular Solution

GCD condition is true. The next step is to find a specific solution to the equation. The extended Euclidean algorithm is one way to do this. This algorithm not only finds the GCD of two numbers but also provides integers x₀ and y₀ to satisfy ax + by = GCD(a, b). So, we adjust x₀ and y₀ to fit ax + by = c.

Example

Let us solve 7x + 11y = 35.

GCD Condition

The GCD of 7 and 11 is 1, which divides 35, so a solution exists.

Extended Euclidean Algorithm

We use it to find numbers x₀ and y₀ such that 7x₀ + 11y₀ = 1. After some calculations, we will get x₀ = 8 and y₀ = −5. This is satisfying 7 × 8 + 11 × (−5) = 1.

Adjust for c − To make the equation equal 35 instead of 1, we must multiply both x₀ and y₀ by 35.

So, x = 8 × 35 = 280 and y = −5 × 35 = −175 and it is a particular solution.

Finding the General Solution

After having the particular solution (x₀, y₀), we can use it to generate all possible solutions to the equation.

If (x₀, y₀) is one solution, then the general solution is given by −

$$\mathrm{x \:=\: x_0 \:+\: \frac{b}{\text{GCD}(a,b)} \:\times\: k}$$

$$\mathrm{y \:=\: y_0 \:+\: \frac{a}{\text{GCD}(a,b)} \:\times\: k}$$

where k is an integer that can take any value.

Example

Using our earlier example 7x + 11y = 35, with the particular solution x = 280 and y = −175, the general solution will be −

$$\mathrm{x \:=\: 280 \:+\: 11k}$$

$$\mathrm{y \:=\: -175 \:-\: 7k}$$

For any integer k, (x, y) will be a solution to 7x + 11y = 35.

If we take k = 0 we get the solution (280, −175). If k = 1 (291, −182) and so on. This gives us a full range of integer solutions to the equation.

Applications of Linear Diophantine Equations

We have understood how to solve these equations. The Linear Diophantine equations have a wide range of applications because they work on integer solutions.

• Currency and Change Problems − Suppose we need to make exactly 27 using only 5 and 8. This can be modeled as the equation 5x + 8y = 27. By solving this, we can find how many of each bill is needed or determine if it is even possible.
• Inventory and Packaging Problems − Imagine we are packing items into crates. Now each crate holds a fixed number of items. If we want an exact total, a Diophantine equation can tell us how many of each type of crate is needed.

Special Cases and No-Solution Situations

Sometimes, a Diophantine equation might not have any solution. For instance, if the GCD of a and b does not divide c. So, it does not have any integer solutions the equation.

For example, consider 4x + 6y = 13. Here, the GCD of 4 and 6 is 2. This is not divide 13. So, we know there are no integer solutions.

Conclusion

In this chapter, we explored the concept of solving Linear Diophantine equations in discrete mathematics. We began by understanding the structure of these equations and the conditions for solutions based on the GCD criterion.

We then walked through the steps to find a particular solution, discussed how to generate the general solution, and examined real-world applications. Additionally, we covered cases where no solutions exist.

Linear Diophantine equations provide a straightforward approach to solving integer-based problems and have numerous applications in discrete mathematics.