1. Finding the Greatest Common Factor of Two Numbers must be common to both numbers. We We are looking for a factor. The factor need to pick the greatest of such common factors.

2. Method 1 The GCF of 36 and 90 1) List the factors of each number. 36: 1 2 3 4 6 36 18 24 9 2) Circle the common factors. 90: 1 2 3 5 6 9 90 45 30 18 15 10 3) The greatest of these will be your Greatest Common Factor: 18

3. Method 2 The GCF of 36 and 90 1) Prime factor each number. 36 = 2 ● 2 ● 3 ● 3 2) Circle each pair of common prime factors. 90 = 2 ● 3 ● 3 ● 5 3) The product of these common prime factors will be 2 ● 3 ● 3 = 18 the Greatest Common Factor:

4. Finding the Least Common Multiple of Two Numbers must be common to both numbers. We We are looking for a multiple. The multiple need to pick the least of such common multiples.

5. Method 1 The LCM of 12 and 15 1) List the first few multiples of each number. 12: 12 24 36 48 60 72 84 90 108 120 2) Circle the common multiples. 15: 15 30 45 60 75 90 105 120 135 3) The least of these will be your Least Common Multiple: 60

6. Method 2 The LCM of 12 and 15. 1) Prime factor each number. 12 = 2 ● 2 ● 3 2) Circle each pair of common prime factors. 15 = 5 ● 3 4) Multiply together one factor from each circle to get the 3 ● 2 ● 2 ● 5 = 60 Least Common Multiple : 3) Circle each remaining prime factor. Note that the common factor, 3, was only used once.

7. Method 3: Find both GCF and LCM at Once. 1) Make the following table. 7290 The GCF and LCM of 72 and 90 2) Divide each number by a common factor. 3) Divide the new numbers by a common factor. Repeat this process until there is no longer a common factor. 9 8 10 2 4 5 The product of the factors on the left is the GCF: 9 ● 2 = 18 The product of the factors on the left AND bottom is the LCM: 9 ● 2 ● 4 ● 5 = 360

8. Method 3: Find both GCF and LCM at Once. 1) Make the following table. 96144 One more example: The GCF and LCM of 96 and 144 2) Divide each number by a common factor. 3) Divide the new numbers by a common factor. 4) Repeat this process until there is no longer a common factor. 2 48 72 6 8 12 The product of the factors on the left is the GCF: 2 ● 6 ● 4 = 48 The product of the factors on the left AND bottom is the LCM: 2 ● 6 ● 4 ● 2 ● 3 = 288 4 23 Note that you can pick any common factor to start and any remaining common factor for each step. Try starting by dividing by 3 to see that this is so.

9. Work on this problem: Juan, Sean and Jane are night guards at an industrial complex. Each starts work at the central gate at 12 midnight. Each guard spends the night repeating a round which starts and ends at the gate. Juan’s round takes 30 minutes; Sean’s round takes 40 minutes; and Jane’s round takes 80 minutes. If they all head out from the gate at midnight, what is the next time that they will all be at the gate.

10. Juan, will return at 12:30, 1:00, 1:30 and so forth. Sean, will return at 12:40, 1:20, 2:00 and so forth. Jane, will return at 1:20, 2:40, 4:20 and so forth. Working with times can be awkward. It is best to work with minutes. Juan, will return after 30 minutes, 60 minutes, 90 minutes, and so forth. Sean, will return after 40 minutes, 80 minutes, 120 minutes, and so forth. Jane, will return after 80 minutes, 160 minutes, 240 minutes, and so forth.

11. You should recognize this as an application of the Least Common Multiple. Sean: 40, 80, 120, 160, 200, 240, 280, 320, … Juan: 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, … Jane: 80, 160, 240, 320, … After 240 minutes they are all at the gate.

12. You can also model the rounds this way. 30 minutes 40 minutes 80 minutes Juan Sean Jane After four rounds for Juan and three rounds for Sean, they are both back at the gate. Every time Jane comes back to the gate, Sean is there. It is only in 240 minutes, after Juan has made 8 rounds, Sean has made 6 rounds and Jane has made 3 rounds, that all three meet at the gate.

13. What have we forgotten? We know that the guards meet at the gate again after 240 minutes, however the problem asks for a time. 240 minutes divided by the 60 minutes in an hour give us 4 hours. 4 hours after 12 midnight is 4 a.m. Juan, Sean and Jane are night guards at an industrial complex. Each starts work at the central gate at 12 midnight. Each guard spends the night repeating a round which starts and ends at the gate. Juan’s round takes 30 minutes; Sean’s round takes 40 minutes; and Jane’s round takes 80 minutes. If they all head out from the gate at midnight, what is the next time that they will all be at the gate. The guards meet at the gate again at 4 a.m.

14. Now work on this problem: You neighbor is putting down a floor with rectangular pieces of plywood. Each piece of plywood is 6 feet by 8 feet. If the floor is square, what is the least possible number of plywood pieces used? Draw a diagram of the situation and solve.

15. 8’ 6’ 8’ 16’ 6’ 12’ 24’ 18’ 24’ Start with one 6 x 8 board and add boards to the right and below until you have a square. You will need to click to add boards. We have our square floor. It is 24 feet by 24 feet. It uses 4 x 3 = 12 boards. The area of the floor is 24 x 24 = 576 square feet.

16. You neighbor is putting down a floor with rectangular pieces of plywood. Each piece of plywood is 6 feet by 8 feet. If the floor is square, what is the least possible number of plywood pieces used? Draw a diagram of the situation and solve. Reread the problem to remember what it asked us to find. We need to find the minimum number of boards that will make a square floor: 12 boards are needed to make a square floor.