Quadratic Equations
Integer Solutions of Exponential-Quadratic Equation
nta_pyq_2024_jan
Grade 11
Question:
Let the set $C=\{(x,y)\mid x^2-2^y=2023,\,x,y\in\mathbb{N}\}$. Then $\displaystyle\sum_{(x,y)\in C}(x+y)$ is equal to
Step-by-Step Solution
Key Concept: $x^2-2^y=2023$. $x^2=2023+2^y$. For $y=1$: $x^2=2025=45^2\Rightarrow x=45$. Check other $y$: $y=2$: $x^2=2027$ (not perfect square); $y=3$: $2031$; ... only $(x,y)=(45,1)$ works for natural numbers.
$(x,y)=(45,1)$ is the only solution. $\sum(x+y)=46$.
Correct Answer: 46