# Factors & Products

## Definition of Factor Pairs

There are many ways to count how many tiles there are in the picture, but since the tiles are arranged in an array, an equal number in each row, the most efficient way to count them is with multiplication. There are 3 rows with 3 tiles in each row. Each 3 represents a **factor**. To find the total, multiply the **factor pairs** 3 and 3, which equals 9 tiles in all. Factors are numbers multiplied to result in a total. Since a pair refers to two items, a factor pair is two factors.

### Product of Two Factors

When two numbers are multiplied, the answer they produce is called the **product**. In the tile example, 3 x 3 = 9. The product of 3 x 3 is 9 and 9 is **divisible** by 3.

The order of the factor pairs in a multiplication equation does not change the product. Factors can be multiplied forward and backward.

### Examples of Factor Pairs

Products can have more than one factor pair. For example, the number 6 has two factor pairs because 6 can be composed in different ways. The following arrays show various ways to make 6. The numbers of **rows** and **columns** are factors.

The factors of 6 can be expressed in a list as 1, 2, 3, and 6 or as pairs 1 and 6 and 2 and 3. Since multiplication is commutative, the same forward and backward, it is not necessary to repeat factor pairs in reverse order. If 2 and 3 are a factor pair of 6, it is assumed that 3 and 2 are also a factor pair of 6.

Another number with more than one factor pair is 24. To find factor pairs for 24, think about the ways 24 can be divided evenly without remainders.

{eq}24 \div 6 = 4 {/eq} and {eq}24 \div 4 = 6 {/eq}

{eq}24 \div 2 = 12 {/eq} and {eq}24 \div 12 = 2 {/eq}

{eq}24 \div 3 = 8 {/eq} and {eq}24 \div 8 = 3 {/eq}

{eq}24 \div 1 = 24 {/eq} and {eq}24 \div 24 = 1 {/eq}

So, the factor pairs for 24 are 1 and 24, 2 and 12, 6 and 4, and 3 and 8. Or, listed out the factors of 24 are 1, 2, 3, 4, 6, 8, and 12.

Here are a few more factor pairs.

15: 1 and 15, 3 and 5

20: 1 and 20, 2 and 10, 4 and 5

27: 1 and 27, 3 and 9

## Definition of a Factor Pair

Let's say a mother in your neighborhood hires you to babysit for $75 a night, and you can babysit four nights of the week. Of course, to figure out your weekly income, we would multiply the nightly income by the number of nights you can work:

75 * 4 = 300

You'll make $300 in one week! Not bad.

**Factors** are the numbers you multiply together. In this multiplication sentence, 75 and 4 are the factors. 300 is the **product**, which is the answer we get when we multiply two factors together.

So, in this multiplication sequence, 3 * 4 = 12, 3 and 4 are the factors, and 12 is the product.

## What is the Difference Between a Factor and a Multiple?

Factors are numbers that are multiplied with each other, and the answer that results is the product. Another way to describe the product of two factors is to say it is a multiple of a number. Is 9 a multiple of 3? Yes! 9 is a multiple of 3 because when the factors 3 and 3 are multiplied, 9 is the product. **Skip-counting** by 3 is also a way to find out its multiples- 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39... are all multiples of 3. They are products of 3 multiplied by another factor.

Is 17 a multiple of 3? No, it is not. When skip counting by 3, 17 is not a number said. Likewise, there is no factor that can be multiplied by 3 to equal 17.

## Facts of Factor Pairs

- Every number has 1 as one of its factors.
- If a number's
*only*factors are 1 and the number itself, the number is prime. - If a number has more factors than 1 and itself, the number is
**composite**. - Zero is not a factor of any other number besides itself because multiplying a number by 0 will always result in 0. So, the factors of 0 are infinite!
- In a division equation, the divisor and quotient are factors of the dividend if the remainder is 0.

## What are Factor Pairs of 12, 13, 2?

- Factor pairs of 12
- 1 and 12, 2 and 6, 3 and 4

12 is a composite number because it has more factors than 1 and itself.

- Factor pair of 13
- 1 and 13

13 is a prime number because its only factors are 1 and itself.

- Factor pair of 2
- 1 and 2

2 is also a prime number because its only factors are 1 and itself.

## Lesson Summary

**Factors** are numbers multiplied by each other to equal a **product**. Any numbers can be **factor pairs** if they are being multiplied. One way factors can be found is by counting the **rows** and **columns** in an array.

Factors can also be found through **divisibility**. If a number can be divided by another without any remainder, that number is a factor. Every number has 1 as a factor. Some numbers only have 1 factor pair including 1 and itself. Those numbers are prime. Other numbers have more than 1 factor pair and are called **composite numbers**. Multiples are the answers to multiplication problems and can also be found by **skip-counting** by a number.

## Pairing Up

**Factor pairs** are pairs of factors we multiply together to get a certain product. Let's go back to our second multiplication sentence: 3 * 4 = 12.

4 and 3 is one factor pair that can give us a product of 12. But we can also write the following multiplication sentences that give a product of 12:

- 1 * 12 = 12
- 2 * 6 = 12

So, the factor pairs that will give us a product of 12 are 1 and 12, 2 and 6, and 3 and 4.

Generally speaking, the larger a number is, the more factor pairs it will have. This isn't true for prime numbers because their only factor pairs are 1 and the number itself. For example, 937 is a large number, but it's also a prime number. So, its only factor pair is 1 and 937.

## Examples

Let's determine the factor pairs for 40. We can write the following multiplication sentences to get a product of 40:

- 1 * 40 = 40
- 2 * 20 = 40
- 4 * 10 = 40
- 5 * 8 = 40

So, 40's factor pairs are 1 and 40; 2 and 20; 4 and 10; 5 and 8.

Up for a challenge? Let's try a 3-digit number. We'll find the factor pairs for 100:

- 1 * 100 = 100
- 2 * 50 = 100
- 4 * 25 = 100
- 5 * 20 = 100
- 10 * 10 = 100

100's factor pairs are 1 and 100; 2 and 50; 4 and 25; 5 and 20; 10 and 10.

## Zero - A Special Case

What about 0? Let's look at the multiplication sequences that give us a product of 0:

- 0 * 1 = 0
- 0 * 2 = 0
- 0 * 3 = 0
- 0 * 4 = 0
- 0 * 1,000,000 = 0

You get the picture. 0 has an infinite number of factor pairs because any number times 0 equals 0!

## Lesson Summary

In multiplication, the numbers we multiply together are **factors**. The answer we get when we multiply is the **product**. Two numbers that we can multiply together to get a certain product make a **factor pair**.

## Factor Pair Overview

Terms | Explanations |
---|---|

Factors | the numbers you multiply together |

Product | the answer we get when we multiply two factors together |

Factor pairs | pairs of factors we multiply together to get a certain product |

## Learning Outcomes

When this lesson is over, you should be able to:

- Define factors
- Describe a product
- Explain what makes up a factor pair

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## Definition of a Factor Pair

Let's say a mother in your neighborhood hires you to babysit for $75 a night, and you can babysit four nights of the week. Of course, to figure out your weekly income, we would multiply the nightly income by the number of nights you can work:

75 * 4 = 300

You'll make $300 in one week! Not bad.

**Factors** are the numbers you multiply together. In this multiplication sentence, 75 and 4 are the factors. 300 is the **product**, which is the answer we get when we multiply two factors together.

So, in this multiplication sequence, 3 * 4 = 12, 3 and 4 are the factors, and 12 is the product.

## Pairing Up

**Factor pairs** are pairs of factors we multiply together to get a certain product. Let's go back to our second multiplication sentence: 3 * 4 = 12.

4 and 3 is one factor pair that can give us a product of 12. But we can also write the following multiplication sentences that give a product of 12:

- 1 * 12 = 12
- 2 * 6 = 12

So, the factor pairs that will give us a product of 12 are 1 and 12, 2 and 6, and 3 and 4.

Generally speaking, the larger a number is, the more factor pairs it will have. This isn't true for prime numbers because their only factor pairs are 1 and the number itself. For example, 937 is a large number, but it's also a prime number. So, its only factor pair is 1 and 937.

## Examples

Let's determine the factor pairs for 40. We can write the following multiplication sentences to get a product of 40:

- 1 * 40 = 40
- 2 * 20 = 40
- 4 * 10 = 40
- 5 * 8 = 40

So, 40's factor pairs are 1 and 40; 2 and 20; 4 and 10; 5 and 8.

Up for a challenge? Let's try a 3-digit number. We'll find the factor pairs for 100:

- 1 * 100 = 100
- 2 * 50 = 100
- 4 * 25 = 100
- 5 * 20 = 100
- 10 * 10 = 100

100's factor pairs are 1 and 100; 2 and 50; 4 and 25; 5 and 20; 10 and 10.

## Zero - A Special Case

What about 0? Let's look at the multiplication sequences that give us a product of 0:

- 0 * 1 = 0
- 0 * 2 = 0
- 0 * 3 = 0
- 0 * 4 = 0
- 0 * 1,000,000 = 0

You get the picture. 0 has an infinite number of factor pairs because any number times 0 equals 0!

## Lesson Summary

In multiplication, the numbers we multiply together are **factors**. The answer we get when we multiply is the **product**. Two numbers that we can multiply together to get a certain product make a **factor pair**.

## Factor Pair Overview

Terms | Explanations |
---|---|

Factors | the numbers you multiply together |

Product | the answer we get when we multiply two factors together |

Factor pairs | pairs of factors we multiply together to get a certain product |

## Learning Outcomes

When this lesson is over, you should be able to:

- Define factors
- Describe a product
- Explain what makes up a factor pair

To unlock this lesson you must be a Study.com Member.

Create your account

#### What are factor pairs of 24?

There are several factors of 24. The factor pairs are 1 and 24, 2 and 12, 3 and 4, and 4 and 6.

#### What are the factor pairs of 12?

The factor pairs of 12 are 1 and 12, 2 and 6, and 3 and 4. 12 can be divided by any of these factors without a remainder.

#### What is a factor pair for 2?

2 has only one factor pair- 1 and 2. Since 2 has no other factor pairs, it is a prime number.

#### What are factors and factor pairs?

Factors are numbers that are multiplied together to equal a product. Factor pairs are the two numbers that are multiplied.

#### How do you find the factor pairs?

You can find factor pairs by creating an array for the total. The number of rows and columns are a factor pair. You can also use division to find factor pairs. When a number is divided by another, the divisor and quotient are factors, as long as the remainder is 0.

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