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Greatest Common Factor Using Lists

GCF is the product of identical prime factors

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Greatest Common Factor Using Lists

Let’s Think About It

Mara is in making flower arrangements for a party. She has 48 carnations and 42 daisies. She wants there to be an equal number of carnations and daisies in each bouquet. What is the most number of bouquets she can make? How many of each flower will they contain?

In this concept, you will learn to find the greatest common factors of numbers using lists.

Guidance

The greatest common factor (GCF) is the greatest factor that two or more numbers have in common. One way to find the GCF is to make lists of the factors for two numbers and then choose the greatest factor that the two factors have in common.

Find the GCF for 12 and 16. It is helpful to order them from smallest to largest in order to make sure that you cover every factor.

First, find all the factors of 12 and 16 and write them in a list in the order of least to greatest.

\begin{align*}\begin{array}{rcl} && 12 - 1, 2, 3, 4, 6, 12\\ \\ && 16 - 1, 2, 4, 8, 16 \end{array}\end{align*}

One way to check if all the factors are listed is to use the rainbow method. Draw a line from one part of a factor pair to the other. The resulting image should resemble a rainbow.

License: CC BY-NC 3.0

Next, identify the GCF, the largest number that appears in both lists. The GCF for 12 and 16 is 4.

Guided Practice

What is the GCF of 140 and 124?

First, find all the factors of 140 and 124 and write them in a list in the order of least to greatest.

\begin{align*}\begin{array}{rcl} && 140 - 1, 2, \underline{4}, 5, 7, 10, 14, 20, 28, 35, 70, 140\\ \\ && 124 - 1, 2, \underline{4}, 31, 62, 124 \end{array}\end{align*}

Next, identify the GCF, the largest number that appears in both lists. The GCF for 140 and 124 is 4.

Examples

Find the GCF for each pair of numbers.

Example 1

\begin{align*}24 \text{ and } 36\end{align*}

First, find all the factors of 24 and 36 and write them in a list in the order of least to greatest.

\begin{align*}\begin{array}{rcl} && 24 - 1, 2, 3, 4, 6, 8, \underline{12}, 24\\ \\ && 36 - 1, 2, 3, 4, 6, 9, \underline{12}, 18, 36 \end{array}\end{align*}

Next, identify the GCF, the largest number that appears in both lists. The GCF for 24 and 36 is 12.

Example 2

\begin{align*}10 \text{ and } 18\end{align*}

First, find all the factors of 10 and 18 and write them in a list in the order of least to greatest.

\begin{align*}\begin{array}{rcl} && 10 - 1, \underline{2}, 5, 10\\ \\ && 18 - 1, \underline{2}, 3, 6, 9, 18 \end{array}\end{align*}

Next, identify the GCF, the largest number that appears in both lists. The GCF for 10 and 18 is 2.

Example 3

\begin{align*}18 \text{ and } 45\end{align*}

First, find all the factors of 18 and 45 and write them in a list in the order of least to greatest.

\begin{align*}\begin{array}{rcl} && 18 - 1, 2, 3, 6, \underline{9}, 18\\ \\ && 45 - 1, 3, 5, \underline{9}, 25, 45 \end{array}\end{align*}

Next, identify the GCF, the largest number that appears in both lists. The GCF for 18 and 45 is 9.

Follow Up

Remember Mara and her flowers?

Mara has 48 carnations and 42 daisies and wants each bouquet to have the same number of flowers. Compare the factors 48 and 42 and find the greatest common factor.

First, find all the factors of 48 and 42 and write them from least to greatest.

\begin{align*}\begin{array}{rcl} && 48 - \mathbf{1}, \mathbf{2}, \mathbf{3}, 4, \mathbf{6}, 8, 12, 16, 24, 48\\ && 42 - \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{6}, 7, 14, 21, 42 \end{array}\end{align*}

Then, identify the GCF. The GCF for 48 and 42 is 6.

Next, find the number of carnations and daisies in 6 bouquets.

\begin{align*}\begin{array}{rcl} && \text{carnations}: \ 48 \div 6 = 8\\ && \text{daisies}: 42 \div 6 = 7 \end{array}\end{align*}

The most number of bouquets Mara can make will be 6. Each will have 8 carnations and 7 daisies.

Video Review

http://www.youtube.com/watch?v=g8QaKFuvAE0

http://www.youtube.com/watch?v=KbBJcdDY_VE

Explore More

Find the GCF for each pair of numbers.

  1. 9 and 21
  2. 4 and 16
  3. 6 and 8
  4. 12 and 22
  5. 24 and 30
  6. 35 and 47
  7. 35 and 50
  8. 44 and 121
  9. 48 and 144
  10. 60 and 75
  11. 21 and 13
  12. 14 and 35
  13. 81 and 36
  14. 90 and 80
  15. 22 and 33
  16. 11 and 13
  17. 15 and 30
  18. 28 and 63
  19. 67 and 14
  20. 18 and 36

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 5.5. 

Vocabulary

factor

factor

Factors are the numbers being multiplied to equal a product. To factor means to rewrite a mathematical expression as a product of factors.
Greatest Common Factor

Greatest Common Factor

The greatest common factor of two numbers is the greatest number that both of the original numbers can be divided by evenly.
Product

Product

The product is the result after two amounts have been multiplied.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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