["# Solving \( a^2 + b^2 = c^2 = 100 \): A Sweet Introduction to Pythagorean Triples and Integer Solutions", "The equation $ a^2 + b^2 = c^2 = 100 $ presents a fascinating intersection of geometry and number theory, inviting exploration into Pythagorean triples—sets of integers $ (a, b, c) $ where $ a^2 + b^2 = c^2 $—and specifically those satisfying $ c^2 = 100 $. In this article, we'll unpack the meaning of this equation, identify valid integer solutions, and explore how it connects to well-known principles in mathematics, particularly the Pythagorean theorem and Diophantine equations.", "---", "## What Does \( a^2 + b^2 = c^2 = 100 \) Mean?", "At first glance, $ a^2 + b^2 = c^2 = 100 $ might seem contradictory because it sets two expressions equal: $ a^2 + b^2 = c^2 $ and $ c^2 = 100 $. Thus, the equation simplifies to:
\n$$
\na^2 + b^2 = 100
\n$$
\nwith the additional constraint that $ c = \sqrt{100} = 10 $. So we are really focused on finding all positive integer pairs $ (a, b) $ such that the sum of their squares equals 100.", "---", "## Understanding Pythagorean Triples", "The classical Pythagorean equation $ a^2 + b^2 = c^2 $ generates the famous Pythagorean triples—sets of whole numbers like $ (3,4,5) $ or $ (5,12,13) $. But here, we fix $ c = 10 $, so only triples where the hypotenuse is exactly 10 matter.", "Because $ c = 10 $, the possible integer values for $ a $ and $ b $ must each satisfy $ 1 \leq a, b \leq 9 $ (since $ a, b < c $). We now test integer values below 10 to see which pairs satisfy $ a^2 + b^2 = 100 $.", "---", "## Finding Integer Solutions to $ a^2 + b^2 = 100 $", "Try small values of $ a $ and solve for $ b $:", "- $ a = 1 $: $ b^2 = 100 - 1 = 99 $ → not a perfect square
\n- $ a = 2 $: $ b^2 = 96 $ → not a square
\n- $ a = 3 $: $ b^2 = 91 $ → no
\n- $ a = 4 $: $ b^2 = 84 $ → no
\n- $ a = 5 $: $ b^2 = 75 $ → no
\n- $ a = 6 $: $ b^2 = 64 $ → $ b = 8 $ ✅
\n- $ a = 7 $: $ b^2 = 51 $ → no
\n- $ a = 8 $: $ b^2 = 36 $ → $ b = 6 $ ✅
\n- $ a = 9 $: $ b^2 = 19 $ → no", "Thus, the only positive integer solutions (up to order) are:", "- $ (a, b, c) = (6, 8, 10) $
\n- $ (a, b, c) = (8, 6, 10) $", "Since $ 8^2 + 6^2 = 64 + 36 = 100 = 10^2 $, both satisfy the equation.", "---", "## Check Why These Are Valid Pythagorean Triples", "These pairs form a scaled version of a primitive triple $ (3,4,5) $. Multiplying each term by 2 gives:", "- $ 2 \cdot 3 = 6 $
\n- $ 2 \cdot 4 = 8 $
\n- $ 2 \cdot 5 = 10 $", "So $ (6,8,10) $ is a non-primitive Pythagorean triple. This is a common and efficient way to generate larger valid triples from well-known smaller ones.", "---", "## General Solutions and Symmetry", "Note that $ a $ and $ b $ can be interchanged—both $ (6,8,10) $ and $ (8,6,10) $ are valid. There are no other integer solutions in positive numbers. Negative values are excluded since side lengths are positive real (or integer) quantities.", "---", "## Applications and Educational Value", "Understanding such equations supports:", "- Geometry: Reinforcing how right triangles obey the Pythagorean theorem.
\n- Number Theory: Exploring Diophantine equations—integer solutions to polynomial equations.
\n- Problem Solving: Teaching pattern recognition and strategic trial (e.g., squaring small integers).", "---", "## Variations and Related Problems", "You can extend exploration by:", "- Solving $ a^2 + b^2 = N $ for other values of $ N $, such as 25, 50, or 65.
\n- Investigating scaled triples (e.g., $ k(a,b,c) $ for $ k = 3 \Rightarrow (18,24,30) $).
\n- Studying triples with larger hypotenuses and Pythagorean quadruples (adding a fourth variable).", "---", "## Conclusion", "The equation $ a^2 + b^2 = c^2 = 100 $ elegantly combines algebra and geometry, revealing a clean, solvable case tied to the theorem and practical number systems. The only positive integer solutions with $ c = 10 $ are $ (6, 8, 10) $ and $ (8, 6, 10) $, both rooted in the classic $ 3,4,5 $ triple, scaled by 2. Whether for homework, exams, or casual learning, this example serves as a gateway to deeper mathematical exploration.", "---", "### Keywords:

PythagoreanTriples #a2+b2=c2 #IntegerSolutions #RightTriangles #DiophantineEquations #MathEducation #Geometry #NumberTheory #PythagoreanTheorem #PythagoreanQuadruples #EducationalArticle", "---", "Looking to learn more? Explore Pythagorean triples, explore geometric proofs, and challenge yourself with integer puzzles!"]