Find the number of ordered pairs $(a, b)$ such that $a^2 + b^2 = 50$
Step-by-Step Solution
Key Concept: Find all integer solutions to $a^2 + b^2 = 50$ by testing divisors and using symmetry
We need $a^2 + b^2 = 50$ where $a, b$ are integers. Starting with $a = 1$: $b^2 = 49$, so $b = ±7$ (2 solutions). For $a = 5$: $b^2 = 25$, so $b = ±5$ (2 solutions). For $a = 7$: $b^2 = 1$, so $b = ±1$ (2 solutions). By symmetry, we also get solutions for $a = -1, -5, -7$ with corresponding values of $b$. Thus the total number of ordered pairs is $2 + 2 + 2 + 2 + 2 + 2 = 12$.
Correct Answer: 12