Greatest Common Divisor (Draft)

1. The Greatest Common Divisor
• Understanding Common Factors in Integer
Sets
• Presented by: (Your Name)
• Subject: Mathematics – Number Theory

2. Objectives
• - Define the Greatest Common Divisor (GCD)
• - Understand how to compute the GCD
• - Explain the significance of GCD in number
theory
• - Use different methods to find the GCD
• - Apply knowledge through exercises

3. Description
• The Greatest Common Divisor (GCD) is the
largest number that divides two or more
integers without leaving a remainder.
• It is used in simplifying fractions, solving
Diophantine equations, and is a foundation of
more advanced mathematical concepts.

4. Definition
• Greatest Common Divisor (GCD):
• The GCD of two integers a and b, denoted as
GCD(a, b), is the largest positive integer that
divides both a and b without a remainder.

5. Importance
• - Helps simplify fractions to lowest terms
• - Basis for algorithms like the Euclidean
Algorithm
• - Used in modular arithmetic and
cryptography
• - Key to understanding number relationships
and factorization

6. Explanation
• Example: Find GCD of 36 and 60
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30,
60
• Common Factors: 1, 2, 3, 4, 6, 12
• GCD(36, 60) = 12

7. More Examples
• 1. GCD(24, 32) → Common factors: 1, 2, 4, 8
→ GCD = 8
• 2. GCD(18, 27) → Common factors: 1, 3, 9 →
GCD = 9
• 3. GCD(100, 45) using Euclidean Algorithm:
• 100 ÷ 45 = 2 remainder 10
• 45 ÷ 10 = 4 remainder 5
• 10 ÷ 5 = 2 remainder 0 → GCD = 5

8. Activity
• Try These:
• 1. Find the GCD of 56 and 72
• 2. Use the Euclidean Algorithm to find
GCD(81, 30)
• 3. What is the GCD of 121 and 88?
• 📌 Challenge: Create your own GCD problem
and solve it.

9. Summary
• - GCD is the largest number that divides two
numbers exactly
• - Useful in simplifying and solving
mathematical problems
• - Can be found using listing or Euclidean
Algorithm