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# Factor Pairs

## All sets of two numbers that multiply to give the product

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Factor Pairs

The 6th grade class is going on a field trip to the state fair. There are a total of 156 students on the trip. The teachers are trying to decide on how to evenly group the students so the groups are not too small or too large. How many different combinations of equal groups can they make with 156 students? What combination would produce a group size of around 15 students per group?

In this concept, you will learn how to identify factor pairs.

### Finding Factor Pairs

A factor is a number or a group of numbers that are multiplied together to make a product. Two factors multiplied together for a product is called a factor pair.

Find the factor pairs for 12.

First, find all the factors for 12 starting with 1. Any number multiplied by 1 is that number. Therefore, one is a factor of every number.

1×12=12\begin{align*}1 \times 12 = 12\end{align*}

After starting with 1, move on to 2, then 3 and so on until you have listed out all of the factors for 12. Knowing the multiplication facts is useful for finding factor pairs.

1×122×63×4===121212\begin{align*}\begin{array}{rcl} 1 \times \underline{12} & = & 12\\ 2 \times \underline{6} & = & 12\\ 3 \times \underline{4} & = & 12 \end{array}\end{align*}

Then, list the factor pairs for 12.

121 and 12,2 and 6,3 and 4\begin{align*}12 - 1 \text{ and } 12, 2 \text{ and } 6, 3 \text{ and } 4\end{align*}

A factor is also a number that will divide evenly into another number, so you can find factor pairs by dividing as well. This is useful when finding the factors of larger numbers.

Find the factor pairs for 72

First, find all the factors for 72. Divide 72 starting with the number 1, and continue on until you have found all of the factors that divide into 72 evenly.

72÷1=7272÷3=2472÷6=1272÷2=3672÷4=1872÷8=9\begin{align*}\begin{array}{rcl} && 72 \div 1 = 72 \qquad 72 \div 3 = 24 \qquad 72 \div 6 = 12\\ && 72 \div 2 = 36 \qquad 72 \div 4 = 18 \qquad 72 \div 8 = 9 \end{array}\end{align*}

Remember that you only need to find one factor pair once. 72÷8=9\begin{align*}72\div 8=9\end{align*} and 72÷9=8\begin{align*}72\div 9=8\end{align*} are in the same fact family.

Then, list the factor pairs for 72.

721 and 71,2 and 36,3 and 24,4 and 18,6 and 12,8 and 9\begin{align*}72 - 1 \text{ and } 71, 2 \text{ and } 36, 3 \text{ and } 24, 4 \text{ and } 18, 6 \text{ and } 12, 8 \text{ and } 9\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about the 6th grade field trip to the State Fair.

The teachers are trying to figure out how to evenly group 156 students. Find all the possible group combinations using factor pairs and the combination that closely produces around 15 students per group.

First, find the factor pairs for 156 students.

1×156=1564×39=1562×76=156  6×26=1563×52=15612×13=156\begin{align*}\begin{array}{rcl} && 1 \times 156 = 156 \qquad 4 \times 39 = 156\\ && 2 \times 76 = 156 \qquad \ \ 6 \times 26 = 156\\ && 3 \times 52 = 156 \qquad 12 \times 13 = 156 \end{array}\end{align*}

Then, find total number of group combinations. There are 6 factor pairs, but there are 2 possible combinations per factor pair. For example, the factor pair 1 and 156 can make 1 group of 156 students or 156 groups of 1 student.

6×2=12\begin{align*}6 \times 2 = 12\end{align*}

There are 12 possible combinations of groups. The factor pair that produces around 15 students is 12 and 13, 12 groups of 13 students or 13 groups of 12 students.

#### Example 2

List the factor pairs of 18.

First, find the factor pairs for 18.

1×182×93×6===181818\begin{align*}\begin{array}{rcl} 1 \times 18 & = & 18\\ 2 \times 9 & = & 18 \\ 3 \times 6 & = & 18 \end{array}\end{align*}

Then, list the factor pairs for 18.

181 and 18,2 and 9,3 and 6\begin{align*}18 - 1 \text{ and } 18, 2 \text{ and } 9, 3 \text{ and } 6\end{align*}

#### Example 3

List the factor pairs for the following number.

\begin{align*}36\end{align*}

First, find the factor pairs for 36.

\begin{align*}\begin{array}{rcl} && 1 \times 36 = 39 \qquad 4 \times 9 = 39\\ && 2 \times 18 = 39 \qquad 6 \times 6 = 39\\ && 3 \times 12 = 39 \end{array}\end{align*}

Then, list the factor pairs for 36.

\begin{align*}36 - 1 \text{ and } 36, 2 \text{ and } 18, 3 \text{ and } 12, 4 \text{ and } 9, 6 \text{ and } 6\end{align*}

#### Example 4

List the factor pairs for the following number.

\begin{align*}24\end{align*}

First, find the factor pairs for 24.

\begin{align*}\begin{array}{rcl} && 1 \times 24 \qquad 3 \times 8\\ && 2 \times 12 \qquad 4 \times 6 \end{array}\end{align*}

Then, list the factor pairs for 24.

\begin{align*}24 - 1 \text{ and } 24, 2 \text{ and } 12, 3 \text{ and } 8, \text{ and } 4 \text{ and } 6\end{align*}

#### Example 5

List the factor pairs for the following number.

\begin{align*}90\end{align*}

First, find the factor pairs for 90.

\begin{align*}\begin{array}{rcl} && 90 \div 1 = 72 \qquad 90 \div 5 = 18\\ && 90 \div 2 = 45 \qquad 90 \div 6 = 15\\ && 90 \div 3 = 30 \qquad 90 \div 9 = 10 \end{array}\end{align*}

Then, list the factor pairs for 90.

\begin{align*}90 - 1 \text{ and } 90, 2 \text{ and } 45, 3 \text{ and } 30, 5 \text{ and } 18, 6 \text{ and } 15, 9 \text{ and } 10\end{align*}

### Review

List the factor pairs for each of the following numbers.

1. 12
2. 10
3. 15
4. 16
5. 56
6. 18
7. 20
8. 22
9. 23
10. 25
11. 27
12. 31
13. 81
14. 48
15. 24
16. 30

To see the Review answers, open this PDF file and look for section 5.1.

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### Vocabulary Language: English

TermDefinition
Factors Factors are numbers or values multiplied to equal a product.