Let the set \( C = \{(x, y) \mid x^2 - 2^y = 2023, x, y \in \mathbb{N}\} \).Then\[\sum_{(x, y) \in C} (x + y)\]is equal to ______.
Correct Answer: 46
Approach Solution - 1
Given the equation: \(x^2 - 2^y = 2023\)
Step 1. By trial, we find that \( x = 45 \) and \( y = 1 \) satisfy the equation, as:
\(45^2 - 2^1 = 2025 - 2 = 2023\)
Step 2. Thus, the only solution in \( C \) is \( (x, y) = (45, 1) \).
Step 3. Calculate \( \sum_{(x, y) \in C} (x + y) \):
\(\sum_{(x, y) \in C} (x + y) = 45 + 1 = 46\)
The Correct Answer is: 46
Approach Solution -2
We are given the equation: \[ x^2 - 2^y = 2023 \] and are asked to find the sum \( \sum_{(x,y) \in C} (x + y) \).
Step 1: Solve the equation
The equation is: \[ x^2 - 2^y = 2023 \] Rearranging for \( x \) and \( y \), we get: \[ x^2 = 2023 + 2^y \] To find integer solutions for \( x \) and \( y \), we trial values for \( y \) to see which one yields a perfect square for \( x^2 \).
Step 2: Testing values of \( y \)
We start by testing small integer values for \( y \): - For \( y = 1 \), we get: \[ x^2 = 2023 + 2^1 = 2023 + 2 = 2025 \] \[ \sqrt{2025} = 45 \quad \Rightarrow \quad x = 45 \] Thus, the solution for \( x \) and \( y \) is \( x = 45 \) and \( y = 1 \).
Step 3: Calculate the sum \( x + y \)
The sum of \( x \) and \( y \) is: \[ x + y = 45 + 1 = 46 \]
Final Answer:
The correct answer is: \[ \boxed{46} \]
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