Integer Solutions How Many Pairs Satisfy 2a^2 + 3b^2 = 35

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Introduction to the Problem

This article delves into an intriguing problem from number theory, specifically focusing on finding integer solutions to the equation 2a^2 + 3b^2 = 35. Our primary goal is to determine the number of ordered pairs (a, b), where a and b are integers, that satisfy this equation. This exploration involves understanding the properties of integers, quadratic equations, and systematic methods for finding solutions. We will meticulously analyze the constraints imposed by the equation and derive all possible integer pairs, offering a comprehensive solution to the problem. This exercise not only enhances our problem-solving skills but also provides a deeper appreciation for the elegance and intricacies of number theory. The approach we take involves a combination of algebraic manipulation, logical deduction, and careful examination of possible values. Each step is crafted to ensure clarity and provide a structured pathway to the final answer. By the end of this discussion, you will have a robust understanding of how to tackle similar problems and a clear methodology for determining integer solutions for such equations. The nature of this problem makes it an excellent example for illustrating how theoretical concepts in mathematics translate into practical problem-solving techniques.

Understanding the Equation and Constraints

The given equation is 2a^2 + 3b^2 = 35, where a and b are integers. To solve this, we need to consider the constraints imposed by the equation. First, note that a and b must be integers, which significantly limits the possible solutions. We can rearrange the equation to isolate terms and analyze potential bounds for a and b. For instance, we can rewrite the equation as 2a^2 = 35 - 3b^2. This form helps us understand that 2a^2 must be non-negative and less than or equal to 35, placing an upper limit on the possible values of a. Similarly, 3b^2 = 35 - 2a^2 implies that 3b^2 is also non-negative and less than or equal to 35, bounding the values of b. These bounds are crucial because they allow us to narrow down the search space for integer solutions. Specifically, we can identify the maximum possible values for a^2 and b^2, and consequently, for a and b. This approach of bounding variables is a common technique in Diophantine equations, where we seek integer solutions. The constraints help us avoid testing infinitely many possibilities, making the problem tractable. Understanding these constraints is pivotal in efficiently solving the equation and finding all valid pairs of (a, b). The interplay between the coefficients 2 and 3, and the constant 35, dictates the specific nature of the solutions, making this problem an interesting case study in number theory.

Determining Possible Values for b

To determine the possible integer values for b, we analyze the equation 2a^2 + 3b^2 = 35. From the equation, we can deduce that 3b^2 ≤ 35, which implies that b^2 ≤ 35/3 ≈ 11.67. Since b is an integer, the possible values for b^2 are 0, 1, 4, and 9. Consequently, the possible integer values for b are -3, -2, -1, 0, 1, 2, and 3. Now we have a limited set of values to test for b. This approach significantly reduces the complexity of the problem because we have transformed an infinite set of possibilities into a manageable finite set. For each of these values of b, we will substitute them back into the original equation and check if the resulting equation has integer solutions for a. This is a systematic way to ensure that we cover all possible solutions without missing any. By understanding the constraints and deriving these possible values for b, we have laid a solid foundation for finding the integer solutions for the given equation. This methodical approach is a cornerstone of solving problems in number theory, where careful analysis and logical deduction are key to uncovering the solutions. The simplicity of this step hides its power; it is a crucial turning point in solving the problem efficiently.

Finding Corresponding Values for a

Now that we have the possible values for b, which are -3, -2, -1, 0, 1, 2, and 3, we can substitute each of these into the equation 2a^2 + 3b^2 = 35 to find the corresponding values for a. This step involves straightforward algebraic manipulation and evaluation. For each b, we solve for a and check if a is an integer. Let's go through each case:

  1. If b = -3 or b = 3: 2a^2 + 3(3^2) = 35 simplifies to 2a^2 + 27 = 35, so 2a^2 = 8, and a^2 = 4. Thus, a can be -2 or 2.
  2. If b = -2 or b = 2: 2a^2 + 3(2^2) = 35 simplifies to 2a^2 + 12 = 35, so 2a^2 = 23. In this case, a^2 = 23/2, which does not yield integer solutions for a.
  3. If b = -1 or b = 1: 2a^2 + 3(1^2) = 35 simplifies to 2a^2 + 3 = 35, so 2a^2 = 32, and a^2 = 16. Thus, a can be -4 or 4.
  4. If b = 0: 2a^2 + 3(0^2) = 35 simplifies to 2a^2 = 35. In this case, a^2 = 35/2, which does not yield integer solutions for a.

By systematically substituting each possible value of b, we have identified the corresponding values of a that satisfy the equation. This method ensures that we exhaust all possibilities and accurately find the integer solutions. The process demonstrates the importance of careful substitution and evaluation in solving Diophantine equations. The results we have obtained are crucial for the final step, where we will count the number of integer pairs (a, b) that solve the equation.

Listing and Counting the Solutions

From the previous step, we have identified the integer pairs (a, b) that satisfy the equation 2a^2 + 3b^2 = 35. Let's list them:

  • When b = -3, a can be -2 or 2, giving us the pairs (-2, -3) and (2, -3).
  • When b = 3, a can be -2 or 2, giving us the pairs (-2, 3) and (2, 3).
  • When b = -1, a can be -4 or 4, giving us the pairs (-4, -1) and (4, -1).
  • When b = 1, a can be -4 or 4, giving us the pairs (-4, 1) and (4, 1).

Therefore, the set of integer pairs (a, b) that satisfy the equation is {(-2, -3), (2, -3), (-2, 3), (2, 3), (-4, -1), (4, -1), (-4, 1), (4, 1)}. Counting these pairs, we find that there are 8 distinct solutions. This methodical listing and counting of solutions is a critical step in ensuring the accuracy of our answer. By clearly presenting each solution, we minimize the chance of errors and provide a comprehensive view of the solution set. The final count of 8 solutions answers the original problem, demonstrating the power of our systematic approach. This step underscores the importance of careful bookkeeping and precise enumeration in mathematical problem-solving. The organized presentation of the solutions highlights the clarity and completeness of our solution process.

Conclusion: The Number of Elements

In conclusion, we have systematically found all integer solutions to the equation 2a^2 + 3b^2 = 35. By analyzing the constraints, determining possible values for b, finding corresponding values for a, and carefully listing the solutions, we have identified 8 distinct integer pairs (a, b) that satisfy the equation. Therefore, the number of elements in the set {(a, b) : 2a^2 + 3b^2 = 35, a, b \in \mathbb{Z}} is 8. This problem exemplifies the elegance and rigor of number theory, where methodical approaches and logical deductions are key to finding solutions. The process we followed can be applied to similar Diophantine equations, providing a versatile strategy for problem-solving. Our journey through this problem has not only provided a specific answer but also reinforced the importance of clear thinking and careful execution in mathematical endeavors. The final answer, 8, is a testament to the effectiveness of our step-by-step method and the power of mathematical reasoning. This exploration serves as a valuable example for students and enthusiasts alike, showcasing how complex problems can be tackled with a structured and analytical mindset.

The correct answer is (c) 8.