The sum of the squares of two consecutive positive integers is 365. What is the product of these two integers?
["The Sum of the Squares of Two Consecutive Positive Integers Equals 365: What Is Their Product?", "Have you ever wondered how two close whole numbers combine through squaring and addition? A classic math puzzle reveals that the sum of the squares of two consecutive positive integers equals 365—and discovering these integers offers a straightforward path to find their product.", "### Understanding Consecutive Squares", "Let the smaller of the two consecutive positive integers be $ n $. Then, the next integer is $ n + 1 $. The sum of their squares is:", "$$\nn^2 + (n + 1)^2 = 365\n$$", "Expanding the expression:", "$$\nn^2 + (n^2 + 2n + 1) = 365\n$$", "$$\n2n^2 + 2n + 1 = 365\n$$", "Subtracting 365 from both sides:", "$$\n2n^2 + 2n - 364 = 0\n$$", "Divide the entire equation by 2:", "$$\nn^2 + n - 182 = 0\n$$", "### Solve the Quadratic Equation", "Use the quadratic formula: $ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $, where $ a = 1 $, $ b = 1 $, $ c = -182 $:", "$$\nn = \frac{-1 \pm \sqrt{1^2 - 4(1)(-182)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 728}}{2} = \frac{-1 \pm \sqrt{729}}{2}\n$$", "Since $ \sqrt{729} = 27 $, we have:", "$$\nn = \frac{-1 + 27}{2} = \frac{26}{2} = 13\n$$", "(We discard the negative root since $ n $ must be a positive integer.)", "Thus, the two consecutive integers are $ 13 $ and $ 14 $.", "### Compute Their Product", "Now, calculate the product:", "$$\n13 \ imes 14 = 182\n$$", "### Why This Puts 365 on Track", "Check that it satisfies the original condition:", "$$\n13^2 + 14^2 = 169 + 196 = 365\n$$", "✅ Perfect! The sum matches.", "### Conclusion", "The sum of the squares of two consecutive positive integers equals 365 when the integers are 13 and 14. Their product is:", "$$\n\boxed{182}\n$$", "This elegant algebraic solution not only solves the puzzle but also illustrates the beauty of quadratic equations in everyday number problems. Whether you're a student tackling math competition problems or someone curious about number patterns, this classic remains a favorite for demonstrating the harmony between algebra and arithmetic.", "---", "Key Takeaways:", "- Setup: $ n^2 + (n+1)^2 = 365 $\n- Solve quadratic $ n^2 + n - 182 = 0 $\n- Found $ n = 13 $, so integers are 13 and 14\n- Product: $ 13 \ imes 14 = 182 $", "Start summing squares today—you’ll unlock more mathematical mysteries!"]
