Teaching Notes 1.11: Finding the Greatest Common Factor
When students are asked to find the greatest common factor (GCF) of two numbers via the
method of prime factorization, confusion may arise over the use of the smallest exponent of a
common base or bases. Because they are finding the ‘‘greatest’’ common factor, some students
mistakenly believe they must use the greatest exponent of a common base or bases.- Explain that the greatest common factor is the largest common factor of two or more
numbers. - Instruct your students to find the greatest common factor of 315 and 135. Work through the
process together. - Ask for volunteers to provide the prime factorization of 315 and 135:
- 315 = 3 × 3 × 5 × 7
- 135 = 3 × 3 × 3 × 5
- Ask your students to list the factors that are common to 315 and 135, and then find the prod-
uct. 3× 3 × 5 =45. The GCF of 315 and 135 is 45. - Explain that prime factorization with exponents can be used to find the GCF. Note that
315 and 135 have a common factor of 3^2. They also have a common factor of 5. Therefore,
32 × 5 =45, which is the GCF. - Review the information and example on the worksheet with your students and emphasize
why 2^2 ×3 is the GCF.
EXTRA HELP:
If there are no common factors listed in the prime factorization of two numbers, the GCF is 1.
Although 1 is not a prime number, it is a factor of every number.ANSWER KEY:
(1) 3 (2) 5 (3) 6 (4) 12 (5) 6 (6) 10 (7) 5 (8) 20 (9) 5 (10) 1 (11) 2 (12) 9
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(Challenge)Disagree. Explanations may vary. One explanation is that 1 is the GCF of any two
prime numbers. But 1 may also be the GCF of two numbers that are composite. For example,
1 is the GCF of 4 and 15. Also, 1 may be the GCF of a prime and composite number. For
example, 1 is the GCF of 3 and 10.
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