Euclidean Algorithm (Draft)

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