Euclidean_Algorithm_Num_Theory_Presentation.pptx
- 1. The Euclidean Algorithm • A Method for Finding the Greatest Common Divisor • Presented by: (Your Name) • Subject: Mathematics – Number Theory
- 2. Objectives • - Define and understand the Euclidean Algorithm • - Use the algorithm to compute the GCD of two integers • - Recognize the steps involved in the algorithm • - Apply the method through practical examples • - Strengthen problem-solving skills in number theory
- 3. Description • The Euclidean Algorithm is a systematic method for computing the Greatest Common Divisor (GCD) of two integers. • It repeatedly applies division and uses remainders to find the GCD efficiently.
- 4. Definition • Euclidean Algorithm: • Given two integers a and b (a > b > 0), repeatedly divide a by b and replace a with b and b with the remainder r until r = 0. • The last non-zero remainder is the GCD.
- 5. Importance • - Faster than listing all common divisors • - Essential in cryptography and computer science • - Forms the basis for advanced number theory methods • - Supports simplification of mathematical operations
- 6. Explanation • Example: Find GCD(252, 105) • 252 ÷ 105 = 2 remainder 42 • 105 ÷ 42 = 2 remainder 21 • 42 ÷ 21 = 2 remainder 0 • → GCD = 21 (last non-zero remainder)
- 7. More Examples • 1. GCD(119, 544) • 544 ÷ 119 = 4 R68 • 119 ÷ 68 = 1 R51 • 68 ÷ 51 = 1 R17 • 51 ÷ 17 = 3 R0 → GCD = 17 • 2. GCD(168, 64) • 168 ÷ 64 = 2 R40 • 64 ÷ 40 = 1 R24
- 8. Activity • Try These: • 1. Use the Euclidean Algorithm to find GCD(96, 36) • 2. Compute GCD(315, 45) • 3. Solve GCD(144, 60) using step-by-step division • 📌 Bonus: Create a pair of numbers and test the algorithm yourself!
- 9. Summary • - Euclidean Algorithm is a reliable way to compute GCD • - Involves repeated division using remainders • - Efficient for large numbers and useful in various applications