Lecture Notes: Greatest Common Factor (GCF) and Least Common Multiple (LCM)

Introduction to L Division

• L Division: A method used to find the GCF and LCM of numbers by dividing them by common factors.
• Example numbers: 65 and 169
  • Both can be divided by 13.
  • GCF is the product of all numbers on the left side of the division.
  • For 65 and 169, the GCF is 13.
• Steps for L Division:
  • Divide by the largest common factor.
  • Stop when no further division is possible.

Example Calculations

Example 1: Finding GCF for 48 and 136

• Step 1: Divide by 2: 48 becomes 24, 136 becomes 68.
• Step 2: Divide by 2 again: 24 becomes 12, 68 becomes 34.
• Step 3: Divide by 2 again: 12 becomes 6, 34 becomes 17.
• Conclusion: GCF is 2 x 2 x 2 = 8.

Example 2: Three Numbers (Question J)

• Numbers: 22, 55, 77
• Factors: 3 and 11
• Result: GCF is 33.

Difference between GCF and LCM

• GCF: Largest common factor of given numbers.
  • Process stops when no numbers can be divided by the same factor.
• LCM: Smallest common multiple of given numbers.
  • Continue division even if only two numbers can be divided by the same factor.

Example: Finding LCM for 30 and 55

• Divide by 5: 30 becomes 6, 55 becomes 11.
• LCM Calculation: Multiply entire L shape.
  • LCM is 5 x 6 x 11 = 330.

Case Study: Four Numbers

• Numbers: 1, 4, 6, 9
• Divisions: Two numbers at a time until no further division is possible.
• Example: Divide 6 and 9 by 3, then 4 and 2 by 2.
• Final LCM: 72

Problem-Solving with Products and Sums

• Find two numbers that multiply to a product and add to a sum.
• Example: Product is 72, sum is 22.
  • Factors are 4 and 18.

Application: Scheduling Problem

• Scenario: Tom, Dick, and Harry have different schedules for days off.
• Objective: Find when all three will have a day off together.
• Calculation: Use LCM of their schedules (8, 10, 12 days).
  • Result: LCM is 120 days.

Closing Remarks

• GCF is useful for simplifying expressions and finding common divisors.
• LCM is essential for scheduling and finding synchronization points.
• Understanding these concepts aids in problem-solving across different mathematical contexts.