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5-2 Greatest Common Factor
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Here are two ways to find the GCF of 54 and 90:
• List all the factors of each number, then
choose the greatest common factor.
Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Common factors: 1, 2, 3, 6, 9, 18. The GCF is 18.
• Find the prime factorization of each number,
then multiply the common factors.
54 = 2 • 3 • 3 • 3
90 = 2 • 3 • 3 • 5
Multiply the common Remember: Every number
factors: 2 • 3 • 3 = 18. has 1 as a factor.
A prime number has exactly
So the GCF is 18.
two factors, itself and 1.
A fraction is in simplest form when its numerator
and denominator have a GCF of 1.
Here are two ways to write 54
90 in simplest form:
Divide the numerator
and denominator by
the GCF.
54 18
90 18
Use prime factorization.
54
90
35
1
1
1
1
1
1
22 •• 33 •• 33 •• 53 35
To write equivalent fractions for 54
90 , multiply or
divide both numerator and denominator by the
same nonzero number.
•2
108
Multiply. 54
90 • 2 180
6
9
Divide. 54
10
90 9
List the factors of each number. Then find the GCF for each pair of numbers.
1, 2, 3, 4, 6, 8, 12, 24
1. 24
1, 2, 3, 6, 9, 18
18
20
6
GCF:
GCF:
1, 2, 3, 6, 7, 14, 21, 42
3. 42
36
14
GCF:
1, 2, 3, 4, 6, 9, 12, 18, 36
1, 2, 4, 5, 10, 20
4
4. 96 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
1, 2, 7, 14
14
Copyright © by William H. Sadlier, Inc. All rights reserved.
2. 36
GCF:
1, 2, 3, 4, 6, 9, 12, 18, 36
12
Find the GCF. Use prime factorization.
5. 30 and 54
6. 27 and 90
7. 14 and 28
30 = 2 • 3 • 5
54 = 2 • 3 • 3 • 3
27 = 3 • 3 • 3
90 = 2 • 3 • 3 • 5
14 = 2 • 7
28 = 2 • 2 • 7
GCF: 2 • 3 = 6
GCF: 9
GCF: 14
8. 35 and 28
9. 17 and 19
10. 43 and 13
35 = 5 • 7
28 = 2 • 2 • 7
17 = 1 • 17
19 = 1 • 19
43 = 1 • 43
13 = 1 • 13
GCF: 7
GCF: 1
GCF: 1
11. 25 and 50
12. 50 and 75
13. 8 and 20
25 = 5 • 5
50 = 2 • 5 • 5
50 = 2 • 5 • 5
75 = 3 • 5 • 5
8=2•2•2
20 = 2 • 2 • 5
GCF: 25
GCF: 25
GCF: 4
Lesson 5-2, pages 110–111.
Chapter 5
125
For More Practice Go To:
Find the GCF. List the factors or use prime factorization.
14. 9, 24, 27
15. 15, 45, 60
GCF: 3
16. 13, 15, 22
GCF: 15
18. 36, 63, 72
GCF: 1
19. 64, 48, 32
GCF: 9
17. 56, 72, 40
GCF: 8
20. 48, 60, 84
GCF: 16
21. 42, 75, 90
GCF: 12
GCF: 3
Write each fraction in simplest form. Use the GCF or prime factorization.
6
22. 18
23. 12
24
646
1846
24. 15
20
12 4 12
24 4 12
5 13
26. 36
45
15 4 5
20 4 5
1
52
27. 35
90
2•2•3•3
3•3•5
4
16 4 4
20 4 4
3
54
28. 42
56
5•7
2•3•3•5
55
25. 16
20
7
5 18
4
55
39
29. 104
2•3•7
2•2•2•7
3 • 13
2 • 2 • 2 • 13
3
54
3
58
Write two equivalent fractions for each given fraction. Possible answers shown.
30. 12
32
31. 25
70
12
32
3
32
32. 112
25
70
24
5 8 5 64
34. 102
316
5
50
102
316
51
204
5 158 5 632
32
112
5 14 5 140
35. 14
21
33. 108
144
16
2
36. 24
90
14
21
2
5 72 5 4
54
3
55
105
5 21 5 210
11
110
55
37. 105
24
90
28
5 3 5 42
108
144
5 56 5 7
12
48
5 45 5 180
38. Jesse says that all these fractions are in
simplest form. Nita says that one is not.
Who is correct?
8 2
5 11 9
39. Sergio says that he can quickly find the
GCF of any two consecutive numbers
without knowing any of the factors of
either number. What is his secret?
9 11 17 12 39
3
9
Nita is right, since 39 5 13.
The GCF of two consecutive numbers is 1.
40. Sometimes a pair of composite numbers has 1 as their only common factor.
Find a value of n that is a composite number less than 72 that makes this
statement true. Explain how you found the number. For example, if the
fraction n is in simplest form, then the GCF of n and 72 is 1.
72
25
Possible response: If n 5 25, then 72 is in simplest form. The prime factorization of the composite
number 25 is 52. The prime factorization of 72 is 23 • 32. The only common factor of the numerator and
denominator is 1, so the fraction is in simplest form.
126
Chapter 5
Copyright © by William H. Sadlier, Inc. All rights reserved.
Solve. Check to justify your answer. Check students’ work.