Math LCM and GCF Tutorial

The Least Common Multiple (LCM

• To find the Least Common Multiple (LCM), you should start by listing out the multiples of both numbers involved in the calculation. These multiples are obtained by multiplying each number by the integers (1, 2, 3, and so on). Continue to list the multiples methodically until you identify a common multiple that appears in both lists. The LCM is defined as the smallest of these common multiples, which serves to efficiently represent a shared dividend for the two original numbers.
• This process involves systematically enumerating the numerical products that result from multiplying each original number by consecutive whole numbers (1, 2, 3, etc.), establishing a comprehensive list.
• The Least Common Multiple (LCM) refers to the smallest integer that is exactly divisible by two or more given numbers, often used in solving problems involving fractions or ratios.

Finding the Greatest Common Factor (GCF)

• To find the Greatest Common Factor (GCF), the first step is to list out the factors of both numbers involved in your calculation. Understanding the factors is crucial, as these are the integers that can be multiplied together in various combinations to produce the original number without leaving any remainder.
• When we refer to factors, we specifically mean those numbers that divide the given number completely, resulting in a whole number rather than a decimal or fraction. This method entails identifying all potential integers that can evenly fit into the number.
• Visualize this process as compiling a comprehensive list of every single number that can be multiplied by another to arrive at the original number, reflecting their multiplicative pairs. This exercise is vital for determining their relationship and demonstrating commonalities.
• Ultimately, the GCF is defined as the largest factor that both numbers share, making it a fundamental concept in number theory and useful in simplifying fractions or solving problems involving ratios.

Example 1: Finding the LCM and GCF of 12 and 9

• LCM:
  • Multiples of 12: 12, 24, 36, 48, 60
  • Multiples of 9: 9, 18, 27, 36, 45
  • The LCM of 12 and 9 is 36.
• GCF:
  • Factors of 12: 1, 2, 3, 4, 6, 12
  • Factors of 9: 1, 3, 9
  • The GCF of 12 and 9 is 3.

Example 2: Finding the LCM and GCF of 20 and 15

• LCM:
  • Multiples of 20: 20, 40, 60, 80, 100
  • Multiples of 15: 15, 30, 45, 60, 75
  • The LCM of 20 and 15 is 60.
• GCF:
  • Factors of 20: 1, 2, 4, 5, 10, 20
  • Factors of 15: 1, 3, 5, 15
  • The GCF of 20 and 15 is 5.