MTH 231

Section 4.3Greatest Common Divisors and Least

Common Multiples

Overview

• We now come to a critical stage in student development.

• It is critical because the two concepts in the section are foundations of fraction arithmetic:

1.Greatest common divisors are used to reduce fractions to lowest terms;

2.Least common multiples are also least common divisors, which are needed to add or subtract fractions.

Greatest Common Divisor

• Let a and b be whole numbers. The greatest natural number d that divides both a and b is called their greatest common divisor (or greatest common factor):

),(),( baGCFbaGCDd

Finding the GCD (or GCF)

Method 1: Intersection of Sets• Good to use when the numbers involved are

small and all the factors of each number are easily written down.

• Can be modeled using Venn Diagrams

An Example

• Let A = {factors of 40} and B = {factors of 36}• Then A = {1, 2, 4, 5, 8, 10, 20, 40} and

B = {1, 2, 3, 4, 6, 9, 12, 18, 36}

The greatest element in the set is 4.

}4,2,1{BA

Using Venn Diagrams

Finding the GCD (or GCF)

Method 2: Prime Factorizations (with and without exponents)

• Utilizes and reinforces prime factorization (listed or using factor trees)

• Exponential form can be used with older students (common prime factors to the smallest exponent)

An Example: Using Prime Factors

Find the GCD of 24 and 40.

24 = 2 x 2 x 2 x 3

40 = 2 x 2 x 2 x 5

GCD(24,40) = 2 x 2 x 2 = 8

Using a Factor Tree

Using Prime Factors with Exponents

24 = 23 x 31

40 = 23 x 51

GCD(24,40) = 23 = 8

An Application of GCD

• Sally’s class collected jeans, shirts, and socks for a clothing drive. They want to fill bags with jeans, shirts, and socks so that each bag has the same number of each item. If the class collected 12 pairs of jeans, 18 shirts, and 24 pairs of socks, what is the largest number of bags the class can make?

Least Common Multiple

• Let a and b be natural numbers. The least natural number m that is a multiple of both a and b is called their least common multiple:

),( baLCMm

Finding the LCM

Method 1: Intersection of Sets• Simple, works well if the two numbers are not

large.• Does not necessarily model well using Venn

Diagrams but can be modeled using the number line.

• Note: for numbers that are co-prime, the LCM is the product of the numbers.

An Example

Finding the LCM

Method 2: Prime Factorization• Use prime factorization to write each number

as a product of prime factors (with or without exponents)

• List all the factors in both numbers. For common factors, use the larger exponent.

• Multiply. This result will be your LCM.

An Example

63 = 32 x 71

120 = 23 x 31 x 51

LCM(63,120) = 23 x 32 x 51 x 71 = 2520

An Application of LCM

• Sally’s class is having a Halloween party. They plan to serve hot dogs. If hot dogs are sold in packages of 10, and hot dog buns are sold in packages of 8, what is the smallest number of buns they should buy so that all the hot dogs and buns are used?

• Sally cleans her closet every 4 days and vacuums her room every 5 days. On November 1, she does both. On what day will she do both again?