**Factors, Multiples and the Distributive Property**

What is a Factor?

A factor is a number that divides evenly into another number.

Find all of the factors of 108. Show your work.

Examples of Factor Problems

Find all of the factors of 24. Make a factor table.

What is a factor table?

To make a factor table, a person must list all of the factors in numerical order.

Solution/Work

Make a factor table.

List all of the factors, which are:

Factor Problem # 2

What is GCF?

Greatest common factor is the largest factor that is common to two or more numbers.

How To Find the GCF

Create two factor tables. Find the common factors; then find the greatest one. If the greatest common factor is 1, then the two numbers are relatively prime.

Solution/Work

Sample Problems on GCF

What is the GCF of 24 and 40? Please show your work.

Solution/Work

The GCF of 24 and 40 is 8. You can figure this out by creating factor tables or dividing.

Factor Table Solution

Therefore, the GCF is 8.

Division Method (Only Works for Two Numbers)

Extension Problem

Find the GCF of THREE numbers: 24, 40 and 48.

Extension Problem Solution

The way to figure this out is by creating 3 factor tables.

Aim: How can we understand

factors,

multiples

and

the Distributive Property?

Factors of 24

More on Greatest Common Factor with Three Numbers

Factors of 40

Factors of 48

As you can see, the greatest common factor is

8.

What is a Multiple?

A multiple is a number that can be divided by another number without a remainder.

Examples of multiples

What are the 1st ten multiples of 3?

What are the multiples of 5?

The 1st ten multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.

What is LCM?

LCM stands for least common multiple. Least common multiple is the least numbers into which two or more numbers will divide evenly.

One Way To Find the LCM

There are many ways to find the least common multiple of a number. The first way, while inefficient is to list out many multiples of each number. Then, find the smallest common number.

Second Way to Find LCM

This procedure involves more steps, but while comparing larger numbers, it is more efficient.

1. Find the prime factorization of each number.

2. List out all of the prime factors.

3. Line the numbers up; the repeating numbers go under each other.

4. Cross out the repeating numbers under each other.

5. Multiply the numbers that are left.

What is Prime Factorization?

Prime factorization is a way to find the "prime factors (or factors that are prime) of a positive integer that divide that integer exactly".

This shows how to find the multiples of 3:

More on Prime Factorization

The way to find prime factorization is by creating a factor tree. For example, if a person wanted to find the prime factorization of 24, they would create a factor tree.

Factor Tree of 24

As you can see, the prime factorization of 24 is 2 X 2 X 2 X 3.

Do Now

Please write what you think the definitions of these terms are.

What is the least common multiple of 6 and 8?

Sample LCM Problem

Solution

The first way to solve this problem is by listing the multiples.

6: 6, 12, 18, 24, 30

8: 8, 16, 24, 32, 40

24 is the smallest common multiple.

Solution # 2

Prime factorization of 6: 2 X 3

Prime factorization of 8: 2 X 2 X 2

2 x 3

2 x 2 x 2

__________

2 X 3 X 2 X 2

= 24, so the LCM is 24.

Extension Problem- LCM of 3 Numbers

What is the least common multiple of 20, 28 and 40?

Solution of Extension Problem

Prime factorization of 20: 2 X 2 X 5

Prime factorization of 28: 2 X 2 X 7

Prime factorization of 40: 2 X 2 X 2 X 5

2 x 2 x 5

2 x 2 x 7

2 x 2 x 5 x 2

____________

2 x 2 x 5 x 7 x 2

= 280: the LCM is 280

What is the Distributive Property?

The distributive property lets you multiply each number in a sum and then add the products. The distributive property also works with division.

More on the Distributive Property

The number outside of the parentheses is multiplied or divided by each of the numbers or variables inside of the parentheses.

TIP: Distributive property is very useful when solving equations.

What is a Prime Number?

A prime number is a number that only has two factors: 1 and the number itself.

How To Do the Distributive Property

Example Problem:

5 (3 + 4) = (5 x 3) + (5 x 4)

= 15 + 20

= 35

While this may not be as efficient while using smaller numbers, using distributive property can help you rewrite an expression as the sum of two common factors. It will also help you multiply larger numbers.

To find the multiples of a number, multiply the number by whole numbers.

What is a Composite Number?

A composite number is a number that has three or more factors.

More Example Problems on The Distributive Property (Multiplication)

1) Use the distributive property to simplify the following expression: (6 x 7) + (6 x 3)= _ (7+3)

2) Use the distributive property to solve the following expression:

42 + 18= (_ x 7) + (_ x 3)

Question # 1 Work/Solution

6 x 7= 42, and 6 x 3= 18

42 + 18= 60.

a (7+ 3 ) = 60

a (10) = 60

6 (10) = 60

Question # 2 Work/Solution

As you can see, a is equal to 6.

The factors of 42 are 6 and 7. The factors of 18 are 6 and 3. Therefore, the final expression is (6 x 7) + (6 x 3)

Extension Problem

Use the distributive property to simplify this expression:

10 (a + 15)

Extension Problem Work/Solution

Distribute ten to each number inside of the parentheses:

10 (a + 15)= (10 x a) + 10 x 15