1. Greatest Common Factor (GCF) and Least Common Multiple (LCM)

2. What are Factors of a number?
The factors of a number are all the numbers that can be divided evenly into the number with no remainder. For instance 5 is a factor of 30 because 30 divided by 5 is 6 (a whole number). When finding factors, we typically focus on whole numbers. Example: What are the factors of 48?
All the factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

3. Group challenge! The factors of 120 are:
First group to bring me all of the factors of the following number in order from smallest to largest gets a bonus point on the next group quiz! Find all factors of 120
The factors of 120 are:
1, 2, 3, 4, 5, 6, 8, 10,12, 15, 20, 24, 30, 40, 60, 120.

4. What are common factors?
Factors that two or more numbers have in common are called common factors of those numbers.
The largest common factor of two or more numbers is called their Greatest Common Factor (GCF).
Example: Find the common factors and GCF of 8, 28, and 12.

5. Find the largest number that divides all three numbers.
Finding a GCF
Factors of 8: 1 2 4 8
Factors of 28: 1 2 4 7 14 28
Factors of 12: 1 2 3 4 6 12
Find the largest number that divides all three numbers.
The GCF is 4.

6. Find the largest number that divides all three numbers.
Finding a GCF
Listing all of the factors of all of the numbers take a lot of time!
Find the factors of the smallest number ONLY
Start with the biggest factor and see if it is a factor of the other numbers, if not go to the next biggest and so on…
Factors of 8: 1 2 4 8
Find the largest number that divides all three numbers.
The GCF is 4.

7. Word problems with gcf
Ms. Jeffrey wants to make dog chew toys out of an old set of sheets. She doesn’t want to waste any material and wants to make the sections of sheet as wide as possible but of all uniform width. One sheet is 72 inches wide and the other is 90 inches wide. How wide should she cut the strips so that they are all of uniform width?
(you need the most that is in both!)
The strips should be 18 inches wide.

8. What happens when there are variables?
Find the GCF of each set: (you need the most that is in both! Or all 3 in this case) a) 24x3y4, 60xy2, and 72x2y2 12xy2 b) 15a2b, 9ab2, and 18ab 3ab

9. With variables:
Find the GCF of each set: (you need the most that is in both! Or all 3 in this case) c) 2p4r, 8p3r2, and 16p2r3 2p2r d) 25x3y5, 50x5y3, and 15x4y4 5x3y3

10. Multiple choice: (error analysis)
Find the Greatest Common Factor: 200p5r3, 80p3r2, and 160p4r5 a) 80p3r2 b) 40p5r5 c) 40p3r2 d) 20p3r2

11. Multiple choice: (error analysis)
Find the Greatest Common Factor. Then give a reason why a student might choose one of the other incorrect responses. 100x6y3, 50x3y, and 30x2y2 a) 10x2y b) 10x6y3 c) 50x2y d) 10xy2

12. What are multiples of numbers?
A multiple is the result of multiplying a number by an integer. A common multiple is a number that is a multiple of two or more numbers. The smallest common multiple of two or more numbers is called their least common multiple (LCM).
Some common multiples of 3 and 4 are 12, 24, 36, 48, etc.
The LCM of 3 and 4 is 12.

13. Find the smallest number that is a multiple of all three.
Finding a LCM:
Find the least common multiple of: 8, 28, 12 8 16 24 32 40 48 56 64 72 80 88 96
104
112
120
128
136
144
152
160
168
176
184
192
Multiples of 8:
Multiples of 28: 28 56 84
112
140
168
196
224 12 24 36 48 60 72 84 96
108
120
132
144
156
168
180
192
Multiples of 12:
Find the smallest number that is a multiple of all three.
The LCM is 168.

14. Least Common Multiple for Algebraic Expressions (LCM)
Find the most in either!
Find the LCM of each set of monomials:
15a2b and 27b3
135a2b3
18x3y4 and 30x2y5
90x3y5

15. In your groups: Find the LCM of each set:
(you need the most that is in either! Or all 3 in this case)
2p4r, 8p3r2, and 16p2r3
16p4r3
b) 25x3y5, 50x5y3, and 15x4y4
150x5y5