Lesson Objectives

- Demonstrate the ability to find the prime factorization of a whole number
- Learn how the terms "factor" and "divisor" are related
- Learn how to find the Greatest Common Factor (GCF) for a group of numbers

## How to Find the Greatest Common Factor (GCF) for a Group of Numbers

As we continue to work with fractions, we will encounter fractions that need to be simplified or reduced to lowest terms.
In order to simplify fractions quickly, we need to learn how to find the greatest common factor (GCF) for a group of numbers.
When studying pre-algebra, the terms GCF, GCD, LCM, and LCD are a common source of confusion. We will learn how to find
the LCM and LCD in a future lesson. For now, we will focus on the GCF and GCD.

So what exactly is the greatest common factor/greatest common divisor? The greatest common factor/greatest common divisor, (abbreviated as GCF/GCD) is the largest factor or divisor that is common to a group of numbers. Let's explore the relationship between the terms "factor" and "divisor" using a few examples:

6 x 7 = 42

6 and 7 are factors

42 is the product

We know that division is the opposite operation of multiplication. Therefore if:

6 x 7 = 42

We can state that:

42 ÷ 6 = 7 and 42 ÷ 7 = 6

We previously learned to label the three parts of a division problem as: dividend, divisor, and quotient. In the problem:

42 ÷ 6 = 7

42 is the dividend

6 is the divisor

7 is the quotient

We can also think about the term "divisor" in a different way. The term divisor can also be used to indicate that the division of one number by another leaves no remainder. When working with the term "divisor" its definition will be based on the context of the situation. We can say that 6 is a divisor of 42 since 42 is divisible by 6. We can also label the 6 as the divisor in the division problem:

42 ÷ 6 = 7

In the first case, we are stating that 42 is divisible by 6, meaning there is no remainder from the division. In the second case, we are simply labeling a part of the division problem.

As another example, in the problem:

10 ÷ 3 = 3 R1

We can label the parts of the division problem as:

10 is the dividend

3 is the divisor

3 is the quotient

1 is the remainder

In this example, 3 is labeled as the divisor in our division problem, however, 3 is not a divisor of 10. This is because 10 is not divisible by 3. This usage of this word can be quite confusing. We just remember in the context of these problems, we are speaking of a number that can divide into another. Each factor of a number will also be a divisor of the number. Since 6 and 7 are factors of 42, they will also be divisors of 42. When a number is a divisor of another, we can show the relationship with the "|" symbol placed between the numbers. The divisor is placed on the left, and the dividend on the right.

8 x 7 = 56

8 and 7 are factors of 56

8 and 7 are also divisors of 56

56 ÷ 8 = 7 and 56 ÷ 7 = 8

7|56 » 7 is a divisor of 56

8|56 » 8 is a divisor of 56

9 x 12 = 108

9 and 12 are factors of 108

108 ÷ 12 = 9 and 108 ÷ 9 = 12

9|108 » 9 is a divisor of 108

12|108 » 12 is a divisor of 108

When asked to find the greatest common factor or the greatest common divisor, it has the same meaning. In this lesson, we will refer to this topic as the GCF. Either way, we are looking for the largest factor or divisor that is common to a group of numbers. In other words, the largest number that each number of the group can be divided by and have no remainder. Finding the GCF is a really simple process.

We can see from our table that 40 and 60 each have 2 and 5 as a factor. The key is building the list with the correct amount of each factor.
We can only list what is common to both. 40 has three factors of 2, but 60 only has two. This means when we build our list, we will include only
two factors of 2. For 5, each number has only one in its prime factorization. This means we will add one factor of 5 to the list:

GCF List » 2 x 2 x 5 = 20

GCF(40, 60) = 20

Let's take a look at a few examples.

Example 1: Find the GCF of 90 and 117

Example 2: Find the GCF of 350 and 550

We can use the same approach when we have more than two numbers. Let's look at an example where we find the GCF of three numbers.

Example 3: Find the GCF of 84, 140, and 168

So what exactly is the greatest common factor/greatest common divisor? The greatest common factor/greatest common divisor, (abbreviated as GCF/GCD) is the largest factor or divisor that is common to a group of numbers. Let's explore the relationship between the terms "factor" and "divisor" using a few examples:

6 x 7 = 42

6 and 7 are factors

42 is the product

We know that division is the opposite operation of multiplication. Therefore if:

6 x 7 = 42

We can state that:

42 ÷ 6 = 7 and 42 ÷ 7 = 6

We previously learned to label the three parts of a division problem as: dividend, divisor, and quotient. In the problem:

42 ÷ 6 = 7

42 is the dividend

6 is the divisor

7 is the quotient

We can also think about the term "divisor" in a different way. The term divisor can also be used to indicate that the division of one number by another leaves no remainder. When working with the term "divisor" its definition will be based on the context of the situation. We can say that 6 is a divisor of 42 since 42 is divisible by 6. We can also label the 6 as the divisor in the division problem:

42 ÷ 6 = 7

In the first case, we are stating that 42 is divisible by 6, meaning there is no remainder from the division. In the second case, we are simply labeling a part of the division problem.

As another example, in the problem:

10 ÷ 3 = 3 R1

We can label the parts of the division problem as:

10 is the dividend

3 is the divisor

3 is the quotient

1 is the remainder

In this example, 3 is labeled as the divisor in our division problem, however, 3 is not a divisor of 10. This is because 10 is not divisible by 3. This usage of this word can be quite confusing. We just remember in the context of these problems, we are speaking of a number that can divide into another. Each factor of a number will also be a divisor of the number. Since 6 and 7 are factors of 42, they will also be divisors of 42. When a number is a divisor of another, we can show the relationship with the "|" symbol placed between the numbers. The divisor is placed on the left, and the dividend on the right.

8 x 7 = 56

8 and 7 are factors of 56

8 and 7 are also divisors of 56

56 ÷ 8 = 7 and 56 ÷ 7 = 8

7|56 » 7 is a divisor of 56

8|56 » 8 is a divisor of 56

9 x 12 = 108

9 and 12 are factors of 108

108 ÷ 12 = 9 and 108 ÷ 9 = 12

9|108 » 9 is a divisor of 108

12|108 » 12 is a divisor of 108

When asked to find the greatest common factor or the greatest common divisor, it has the same meaning. In this lesson, we will refer to this topic as the GCF. Either way, we are looking for the largest factor or divisor that is common to a group of numbers. In other words, the largest number that each number of the group can be divided by and have no remainder. Finding the GCF is a really simple process.

### Finding the GCF

- Find the prime factorization for each number
- Create a list of prime factors that are common to all numbers
- The GCF is the product of the numbers on the list
- Optional - it can be helpful to organize information in a table format

Number | Prime Factors | ||||
---|---|---|---|---|---|

40 | 2 | 2 | 2 | 5 | |

60 | 2 | 2 | 3 | 5 |

Number | Prime Factors | ||||
---|---|---|---|---|---|

40 | 2 | 2 | 2 | 5 | |

60 | 2 | 2 | 3 | 5 |

GCF(40, 60) = 20

Let's take a look at a few examples.

Example 1: Find the GCF of 90 and 117

Number | Prime Factors | ||||
---|---|---|---|---|---|

90 | 2 | 3 | 3 | 5 | |

117 | 3 | 3 | 13 |

- Find the prime factorization of each number
- 90 = 2 x 3 x 3 x 5
- 117 = 3 x 3 x 13
- Create a list of prime factors that are common to all numbers
- The GCF is the product of the numbers on the list
- GCF List » 3 x 3 = 9

Example 2: Find the GCF of 350 and 550

Number | Prime Factors | ||||
---|---|---|---|---|---|

350 | 2 | 5 | 5 | 7 | |

550 | 2 | 5 | 5 | 11 |

- Find the prime factorization of each number
- 350 = 2 x 5 x 5 x 7
- 550 = 2 x 5 x 5 x 11
- Create a list of prime factors that are common to all numbers
- The GCF is the product of the numbers on the list
- GCF List » 2 x 5 x 5 = 50

We can use the same approach when we have more than two numbers. Let's look at an example where we find the GCF of three numbers.

Example 3: Find the GCF of 84, 140, and 168

Number | Prime Factors | |||||
---|---|---|---|---|---|---|

84 | 2 | 2 | 3 | 7 | ||

140 | 2 | 2 | 5 | 7 | ||

168 | 2 | 2 | 2 | 3 | 7 |

- Find the prime factorization of each number
- 84 = 2 x 2 x 3 x 7
- 140 = 2 x 2 x 5 x 7
- 168 = 2 x 2 x 2 x 3 x 7
- Create a list of prime factors that are common to all numbers
- The GCF is the product of the numbers on the list
- GCF List » 2 x 2 x 7 = 28

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