Greatest_Common_Factor powerpoint presentation.ppt

Calculate GCF, GCD and HCF of a set of two or more numbers and see the work using factorization.

Enter 2 or more whole numbers separated by commas or spaces.

The Greatest Common Factor Calculator solution also works as a solution for finding:

Greatest common factor (GCF)
Greatest common denominator (GCD)
Highest common factor (HCF)
Greatest common divisor (GCD)
What is the Greatest Common Factor?
The greatest common factor (GCF or GCD or HCF) of a set of whole numbers is the largest positive integer that divides evenly into all numbers with zero remainder. For example, for the set of numbers 18, 30 and 42 the GCF = 6.

Greatest Common Factor of 0
Any non zero whole number times 0 equals 0 so it is true that every non zero whole number is a factor of 0.

k × 0 = 0 so, 0 ÷ k = 0 for any whole number k.

For example, 5 × 0 = 0 so it is true that 0 ÷ 5 = 0. In this example, 5 and 0 are factors of 0.

GCF(5,0) = 5 and more generally GCF(k,0) = k for any whole number k.

However, GCF(0, 0) is undefined.

How to Find the Greatest Common Factor (GCF)
There are several ways to find the greatest common factor of numbers. The most efficient method you use depends on how many numbers you have, how large they are and what you will do with the result.

Factoring
To find the GCF by factoring, list out all of the factors of each number or find them with a Factors Calculator. The whole number factors are numbers that divide evenly into the number with zero remainder. Given the list of common factors for each number, the GCF is the largest number common to each list.

Example: Find the GCF of 18 and 27
The factors of 18 are 1, 2, 3, 6, 9, 18.

The factors of 27 are 1, 3, 9, 27.

The common factors of 18 and 27 are 1, 3 and 9.

The greatest common factor of 18 and 27 is 9.

Example: Find the GCF of 20, 50 and 120
The factors of 20 are 1, 2, 4, 5, 10, 20.

The factors of 50 are 1, 2, 5, 10, 25, 50.

The factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.

The common factors of 20, 50 and 120 are 1, 2, 5 and 10. (Include only the factors common to all three numbers.)

The greatest common factor of 20, 50 and 120 is 10.

Prime Factorization
To find the GCF by prime factorization, list out all of the prime factors of each number or find them with a Prime Factors Calculator. List the prime factors that are common to each of the original numbers. Include the highest number of occurrences of each prime factor that is common to each original number. Multiply these together to get the GCF.

You will see that as numbers get larger the prime factorization method may be easier than straight factoring.

Example: Find the GCF (18, 27)
The prime factorization of 18 is 2 x 3 x 3 = 18.

The prime factorization of 27 is 3 x 3 x 3 = 27.

The occurrences of common prime factors of 18 and 27 are 3 and 3.

So the greatest common factor of 18 and 27 is 3 x 3 = 9.

Example: Find the GCF (20, 50, 120)
The prime factorization of 20 is 2 x 2 x 5 = 20.

The prime

1. Greatest Common Greatest Common
Factor (GCF)Factor (GCF)

2. Vocabulary:
Greatest Common Factor – the largest factor
that two or more numbers have in common.
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

3. When thinking about finding the Greatest When thinking about finding the Greatest
Common Factor, or the GCF…Common Factor, or the GCF…
THINK BACKWARDSTHINK BACKWARDS
FF……FFind the Factorsind the Factors
CC……CCircle Common Factorsircle Common Factors
GG……GGroup Largest Factorroup Largest Factor
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

4. But if that’s too hard…But if that’s too hard…
Simply THINKSimply THINK
GG……GGreatest (largest)reatest (largest)
CC……CCommon (shared)ommon (shared)
FF……FFactoractor
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

5. Important to Remember…Important to Remember…
There are There are TWOTWO methods for finding methods for finding
the GCF of two or more numbers…the GCF of two or more numbers…
Method 1…Method 1…UseUse Book EndsBook Ends
Method 2…Method 2…UseUse Prime FactorizationPrime Factorization
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

6. Finding the GCF: Method 1 – Book EndsFinding the GCF: Method 1 – Book Ends
Example 1Example 1:: Find the GCF of 24 and 36. Find the GCF of 24 and 36.
Step 1: Find the factors of each number.
Step 2: Circle the common factors of the numbers
Step 3: Group or circle the largest factor they have in common
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

7. Finding the GCF: Method 1 – Book EndsFinding the GCF: Method 1 – Book Ends
Example 1Example 1:: Find the GCF of 24 and 36. Find the GCF of 24 and 36.
Step 1: Find the factors of each number.
Step 2: Circle the common factors of the numbers
Step 3: Group or circle the largest factor they have in common
2424: 1, 2, 3, 4, 6, 8, 12, 24: 1, 2, 3, 4, 6, 8, 12, 24
3636: 1, 2, 3, 4, 6, 9, 12, 18, 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
The GCF of 24 and 36 is 12
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

8. Finding the GCF: Method 2 – Prime FactorizationFinding the GCF: Method 2 – Prime Factorization
Example 1Example 1:: Find the GCF of 24 and 36. Find the GCF of 24 and 36.
Step 1: Find the prime factorization of each number.
Step 2: Find the product of the common prime factors
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

9. Finding the GCF: Method 2 – Prime FactorizationFinding the GCF: Method 2 – Prime Factorization
Example 1Example 1:: Find the GCF of 24 and 36. Find the GCF of 24 and 36.
Step 1: Find the prime factorization of each number.
Step 2: Find the product of the common prime factors
2424 3636
GCF = 12
221212
66
2 · 2 · 2 · 32 · 2 · 2 · 32 · 2 · 3 · 32 · 2 · 3 · 3
22
2233
221212
6622
3333
2424: : 22 · · 22 · 2 · · 2 · 33
3636: : 22 · · 22 · 3 · · 3 · 33
2 · 2 · 3 = 122 · 2 · 3 = 12
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

10. Finding the GCF: Method 1 – Book EndsFinding the GCF: Method 1 – Book Ends
Example 2Example 2:: Find the GCF of 12 and 24. Find the GCF of 12 and 24.
Step 1: Find the factors of each number.
Step 2: Circle the common factors of the numbers
Step 3: Group or circle the largest factor they have in common
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

11. Finding the GCF: Method 1 – Book EndsFinding the GCF: Method 1 – Book Ends
Example 2Example 2:: Find the GCF of 12 and 24. Find the GCF of 12 and 24.
Step 1: Find the factors of each number.
Step 2: Circle the common factors of the numbers
Step 3: Group or circle the largest factor they have in common
1212: : 1, 2, 3, 4, 6, 1, 2, 3, 4, 6, 1212
2424: 1, 2, 3, 4, 6, 8, : 1, 2, 3, 4, 6, 8, 1212, 24, 24
The GCF of 12 and 24 is 12
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

12. Finding the GCF: Method 2 – Prime FactorizationFinding the GCF: Method 2 – Prime Factorization
Example 2Example 2:: Find the GCF of 12 and 24. Find the GCF of 12 and 24.
Step 1: Find the prime factorization of each number.
Step 2: Find the product of the common prime factors
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

13. Finding the GCF: Method 2 – Prime FactorizationFinding the GCF: Method 2 – Prime Factorization
Example 2Example 2:: Find the GCF of 12 and 24. Find the GCF of 12 and 24.
Step 1: Find the prime factorization of each number.
Step 2: Find the product of the common prime factors
1212 2424
GCF = 12
2266
2 · 2 · 32 · 2 · 32 · 2 · 3 · 32 · 2 · 3 · 3
2233
221212
6622
2233
1212: : 22 · · 22 · · 33
2424: : 22 · · 22 · 2 · · 2 · 33
2 · 2 · 3 = 122 · 2 · 3 = 12
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

14. Finding the GCF: Method 1 – Book EndsFinding the GCF: Method 1 – Book Ends
Example 3Example 3:: Find the GCF of 16 and 20. Find the GCF of 16 and 20.
Step 1: Find the factors of each number.
Step 2: Circle the common factors of the numbers
Step 3: Group or circle the largest factor they have in common
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

15. Finding the GCF: Method 1 – Book EndsFinding the GCF: Method 1 – Book Ends
Example 3Example 3:: Find the GCF of 16 and 20. Find the GCF of 16 and 20.
Step 1: Find the factors of each number.
Step 2: Circle the common factors of the numbers
Step 3: Group or circle the largest factor they have in common
1616: 1, 2, : 1, 2, 44, 8, 16, 8, 16
2020: 1, 2, : 1, 2, 44, 5, 10, 20, 5, 10, 20
The GCF of 16 and 20 is 4
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

16. Finding the GCF: Method 2 – Prime FactorizationFinding the GCF: Method 2 – Prime Factorization
Example 3Example 3:: Find the GCF of 16 and 20. Find the GCF of 16 and 20.
Step 1: Find the prime factorization of each number.
Step 2: Find the product of the common prime factors
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

17. Finding the GCF: Method 2 – Prime FactorizationFinding the GCF: Method 2 – Prime Factorization
Example 3Example 3:: Find the GCF of 16 and 20. Find the GCF of 16 and 20.
Step 1: Find the prime factorization of each number.
Step 2: Find the product of the common prime factors
1616 2020
GCF = 4
2288
2 · 2 · 52 · 2 · 5
2244
221010
2255
2020: : 22 · · 22 · 5 · 5
2 · 2 = 42 · 2 = 4
2222
2 · 2 · 2 · 22 · 2 · 2 · 2
1616: : 22 · · 22 · 2 · 2 · 2 · 2
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

18. Important to Remember…Important to Remember…
There are There are TWOTWO methods for finding methods for finding
the GCF of two or more numbers…the GCF of two or more numbers…
Method 1Method 1……UseUse Book EndsBook Ends
Method 2Method 2……UseUse Prime FactorizationPrime Factorization
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

19. Guided Practice ProblemsGuided Practice Problems
DirectionsDirections:: Find the GCF of each set of numbers. Find the GCF of each set of numbers.
1. 9, 12, 301. 9, 12, 30
2. 42, 602. 42, 60
3. 48, 643. 48, 64
4. 40a4. 40a
22
b, 48abb, 48ab
44
Greatest Common Factor (GCF)Greatest Common Factor (GCF)

20. Guided Practice ProblemsGuided Practice Problems
DirectionsDirections:: Find the GCF of each set of numbers. Find the GCF of each set of numbers.
1. 9, 12, 301. 9, 12, 30=> => 33
2. 42, 602. 42, 60=> => 66
3. 48, 643. 48, 64=> => 1616
4. 40a4. 40a
22
b, 48abb, 48ab
44
=> => 8ab8ab
Greatest Common Factor (GCF)Greatest Common Factor (GCF)