Greatest Common Factor and Least Common Multiple

How do you find them?

WARM-UPAt the beginning of the summer, Lauren had a balance

$25 in her bank account. She saved a total of $145 from her summer job, which she deposited into her account. Today Lauren withdrew enough money from her account to cover her $20 athletic facilities fee and her $30 uniform fee. If Lauren made no other withdrawals or deposits, what is her account balance?

Starting today, Lauren will earn $20 each week in allowance, which she plans to deposit into her bank account. If Lauren makes no other withdrawals or deposits, after how many weeks will the balance in her bank account be double what it is now?

The balance of Lauren’s account is $120.

Her account will double in 6 weeks.

Warm-Up1. What is the prime factorization of 48?

2. What is the GCF of 35, 21, and 84?

Prime Factorization of 24:2•2•2•23

The GCF of 35, 21, and 84 is 7.

Warm-Up1. Find the GCF of 28 and 42 using prime

factorization.

2. What is the LCM of 12 and 32?The LCM of 12 and 32 is 96.

28 = 2 2 7 and 42 = 2 3 7

The GCF of 28 and 42 is 14.

What is a factor?

The numbers you multiply together to get a larger number

2 and 3 are both factors of 6. What

are the other factors of 6?

What is a prime number?Numbers that have no other factors but itself

and one!What are the first 6 prime numbers?

2, 3, 5, 7, 11, 13

What is prime factorization?The prime numbers that multiply together to

get the original number.

Use the ladder method!

1. Begin with the lowest prime number and see if it is divisible by the original number.

2. Continue until it is no longer divisible by that number, then move on to the next prime number

3. Continue dividing by prime numbers until the only number left inside the ladder is one

What is the prime factorization of 24?

242122

623 3

1Prime Factorization

of 24:2•2•2•3

What is the prime factorization of 39?

3931313

1

Prime Factorization of 39:3•13

What is the Greatest Common Factor?It is the largest of the common factors

between two numbers.

Method 1: Rainbow!What is the greatest common factor of 16 and

36?

16: 1, 2, 4, 8, 16

36: 1, 2, 3, 4, 6, 9, 12, 18, 36

List all the factors of each of the numbers. Once you either have 2 of the same factors or numbers that are so close there are no

other factors between them, then you’ve

reached the middle of the rainbow.

The greatest common factor or 16 and 36 is

4.

Method 2: Find each number’s prime factorization What is the greatest common factor of 12 and

18?

62

32

3

12 2 18

393

1 13

2•2•3 2•3•3

What do they have in common?

2•3=6 The greatest common factor of 12 and 18 is

6.

What is the greatest common factor of 28 and 35?

Use both methods

28: 1, 2, 4, 7, 14, 28

35: 1, 5, 7, 35

2814

22

71

7

35577

1

2•2•7 5•7

Using the prime factorization method,

the GCF is 7.

Using the rainbow method, the GCF is 7.

Warm-Up

What is a multiple?The number you get when you multiply a

number by another number

What are the multiples of 5?

5, 10, 15, 20, 25, 30…

What is the least common multiple?It is the smallest multiple that is shared

between two numbers.

How to find LCMIt’s simple…Just list the multiples of each

number until you find one that is in common!

What is the LCM of 4 and 7?

4: 4, 8, 12, 16, 20, 24, 28

7: 7, 14, 21, 28

What is the LCM of 12 and 16?

12: 12, 24, 36, 48

16: 16, 32, 48

Which should you use to solve the problem, GCF or LCM?

Two shuttles leave the Hard Rock Hotel to go to Universal Studios at the same time. The Minion Madness shuttle returns to the hotel every 8 minutes. The Incredible Hulk shuttle returns to the hotel every 10 minutes. In how many minutes will Minion Madness and the Incredible Hulk leave the hotel together for the second time?

Use the LCM!

The shuttles will leave the hotel at the same time in 40

minutes.

Which should you use to solve the problem, GCF or LCM?

Mr. Schuester directs two show choirs. One choir has 28 students. The other choir has 36 students. For rehearsals, he wants to divide each chorus into the largest possible equal groups with no students left over. How many students will be in each group?

Use the GCF!

Mr. Schuester can divide each choir into groups of 4.

Warm-Up

Distributive Property!!!Let’s have Tim and Moby gives us the basics. :)

http://www.brainpop.com/math/numbersandoperations/distributiveproperty/

How does the distributive property work?

2(2 + 3)The number outside the

parentheses “jumps” over the parentheses

and multiplies by each number inside! (2 • 2) + (2 • 3)

(4) + (6)

=10Just bring down the

sign!

Multiply and then add!

You can write a number or sum in distributive form…

How can you write an expression equal to 33 + 77 using distributive form?

1. Find the GCF of the two numbers. (This will go outside of the parentheses.)

GCF: 11 11()

2. Now use the other factor that multiplies with the GCF to get the original number. (Those 2 numbers go inside of the parentheses.)

3 • 11 & 7 • 11 11(3 + 7)

Now you try one! The number 108 can be expressed as the sum

100 + 8. How can you use the distributive property to rewrite that sum as a multiple of a sum whose addends have no common factors? (Tip: Ignore all the crazy wording and just figure out what’s important!)

4(25 + 2)