Solving for Integral Pairs (X, Y) in a Linear Equation

When dealing with linear Diophantine equations, it is essential to understand how to determine the number of integral pairs (X, Y) that satisfy a given equation. In this article, we explore the equation 141x 517y 4158, and we determine how many integral pairs (X, Y) where 0 ≤ X, Y ≤ 1000 satisfy it. We'll break down the process step-by-step, providing a comprehensive analysis to help SEO optimization.

Introduction to the Problem

The equation in question is a linear Diophantine equation of the form ax by c, where a, b, and c are integers. In this case, a 141, b 517, and c 4158. Our goal is to find the number of integral pairs (X, Y) that satisfy this equation, with the constraint that 0 ≤ X, Y ≤ 1000.

Maximum Values for X and Y

To begin, we need to determine the maximum possible values for X and Y that satisfy the equation under the given constraints. Let's analyze the equation step-by-step:

Case 1: When Y 0

If we set Y 0, the equation simplifies to:

[ 141x 4158 ]

Solving for X, we get:

[ x_{max} leftlfloor frac{4158}{141} rightrfloor leftlfloor 29.485 rightrfloor 29 ]

Case 2: When X 0

Similarly, if we set X 0, the equation simplifies to:

[ 517y 4158 ]

Solving for Y, we get:

[ y_{max} leftlfloor frac{4158}{517} rightrfloor leftlfloor 8.044 rightrfloor 8 ]

Checking for Integral Solutions

Now that we have the maximum values for X and Y, let's check if there are any integral pairs (X, Y) that satisfy the equation 141x 517y 4158 within the given range 0 ≤ X, Y ≤ 1000.

Analysis of Y Values

Since we determined that the possible values for Y are 0, 1, 2, 3, 4, 5, 6, and 7 (up to and including 8), let's substitute these values into the equation and check if X becomes an integer:

For Y 0: For Y 1: For Y 2: For Y 3: For Y 4: For Y 5: For Y 6: For Y 7: For Y 8:

In each case, we need to check if the resulting X value is an integer:

The equation can be rearranged to express X in terms of Y:

[ X frac{4158 - 517Y}{141} ]

For each Y value, X must be an integer.

Verification

Let's verify the calculations for a few Y values to ensure accuracy:

For Y 0:

[ X frac{4158 - 517 cdot 0}{141} frac{4158}{141} 29.485 ] - Not an integer.

For Y 1:

[ X frac{4158 - 517 cdot 1}{141} frac{3641}{141} 25.845 ] - Not an integer.

For Y 2:

[ X frac{4158 - 517 cdot 2}{141} frac{3124}{141} 22.163 ] - Not an integer.

Performing similar calculations for Y 3, 4, 5, 6, and 7, we find that in each case, X does not turn out to be an integer.

Finally, for Y 8:

[ X frac{4158 - 517 cdot 8}{141} frac{2726}{141} 19.324 ] - Not an integer.

Conclusion

After checking all possible integer values for Y (0 to 8), we find that none of them result in an integer value for X. Therefore, the number of integral solutions (X, Y) that satisfy the equation 141x 517y 4158 within the given constraints is zero.

Final Thought on Diophantine Equations

Diophantine equations can be complex to solve, especially when looking for integral solutions. In this problem, we learned the method of checking the constraints and verifying the integer nature of variables. The process is crucial for understanding linear Diophantine equations and can be applied in various mathematical and practical contexts.

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