Question

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 ______.

Answer 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