Session 6, Part A:
Models for Multiples and Factors (75 minutes)

In This Part: The Venn Diagram Model | Finding Prime Factors | The Area Model

 The numbers 24 and 36 have certain things in common, including many common factors -- numbers that divide evenly into both of them. For example 2, 3, and 6 are all common factors. The largest such number is called the "greatest common factor." In this case, the greatest common factor of 24 and 36 is 12. No number greater than 12 is a factor of each of these numbers. Note 2 Another characteristic numbers can share is a common multiple -- a third number that is evenly divisible by both 24 and 36. The smallest such number is called the "least common multiple." In this case, the least common multiple is 72. No number less than 72 is evenly divisible by each number. One way to explore the common factors and multiples of the two numbers is to use a Venn diagram: The circle on the left contains all the prime factors (i.e., counting numbers that have exactly two factors: themselves and 1) of 24, and the circle on the right contains all the prime factors of 36. (The number 1 doesn't qualify as prime, because it has only one factor.) The numbers contained in the intersection are those factors that are in both numbers; i.e., their common factors. That means that the 2s and the 3 in the intersection, both separately and multiplied together (2 • 2, 2 • 3, and 2 • 2 • 3 -- or 2, 3, 4, 6, and 12), are all common factors. Note that the largest of these factors is 12. The greatest common factor (GCF) of 24 and 36 is 2 • 2 • 3, or 12, the product of all the numbers in the overlap. Since the circle on the left contains all the factors of 24, every multiple of 24 must contain all of these factors. Likewise, since the circle on the right contains all the factors of 36, every multiple of 36 must contain all of these factors. In order to be a multiple of both numbers, a number must contain all the factors of both numbers. The smallest number to do this is 2 • 2 • 2 • 3 • 3, or 72, the product of all the factors in the circles. Thus, the least common multiple (LCM) of 24 and 36 is 72. Note 3 Problem A1 Use a Venn diagram to determine the GCF and LCM for 18 and 30.

 Session 6: Index | Notes | Solutions | Video