Download Greatest Common Factor

Number Theory
and Systems
Teacher Key
Greatest Common Factor
Venn Diagram
5
3
5
A
2
3
B
Venn Diagrams are used to illustrate similarities and differences between two or more
objects. They can be used in math to represent similarities and differences between
numbers as well. Each circle represents a composite number. Circle A represents the
number 75 and includes the prime factors 3 and 5. Circle B represents the number 90 and
includes the prime factors 2, 3, and 5.
The numbers 3 and 5 in the center represent common factors of 75 and 90. The greatest
common factor of 75 and 90 is the product of the common prime factors, 15.
Refer to the Venn Diagram below to answer the following questions.
2
5
3
2
3
5
A. What are the two composite numbers?
The two composite numbers are 300 and 450.
B. What is the prime factorization of each number?
The prime factorization of 300 is 2 x 2 x 3 x 5 x 5.
The prime factorization of 450 is 2 x 3 x 3 x 5 x 5.
© 2003 CompassLearning, Inc.
Activity 67115
Number Theory
and Systems
Teacher Key
Greatest Common Factor
C. How did you determine the composite numbers?
You determine the composite numbers by multiplying the prime factors of
each composite number together.
D. What is the greatest common factor of the two numbers?
The greatest common factor of 300 and 450 is 150.
E. How did you use the Venn Diagram to find the greatest common factor?
The greatest common factor can be found by multiplying the common prime
factors found in the overlapping section of the two circles.
F. How would the Venn Diagram look if 250 were added as a third circle?
2
3
3
2
5
5
5
G. What is the greatest common factor among the three numbers?
The greatest common factor among 250, 300, and 450 is 50.
H. List all of the common factors of 250, 300, and 450. How can you use the Venn
diagram to find the common factors?
The common factors are 2, 5, 10, 25, and 50. You can use the Venn diagram to
find the common factors by multiplying all the combinations of the prime
factors that appear in the overlapping section.
© 2003 CompassLearning, Inc.
Activity 67115
Number Theory
and Systems
Teacher Key
Greatest Common Factor
Solve.
I. The greatest common factor of two numbers is 30. Their least common multiple is 420.
One of the numbers is 210. What is the other number?
The other number is 60.
J. Denise is thinking of two numbers. Their greatest common factor is 6. Their least
common multiple is 36. One of the numbers is 12. What is the other number?
The other number is 18.
© 2003 CompassLearning, Inc.
Activity 67115
Number Theory
and Systems
Teacher Key
Connections
Think About It
How can you use the greatest common factor to find the simplest form of a fraction?
SAMPLE RESPONSE: If you divide the numerator and denominator of a fraction by
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their greatest common factor, the resulting equivalent fraction will be in simplest
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form.
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A Floral Arrangement
A florist receives 45 daisies and 60 lilies. The florist charges $1 per daisy and $2 per lily.
What are the different numbers of vases that will allow an equal distribution of flowers and
how much will each cost?
(Assume that the price of each vase is only based on the number and type of flowers
used.)
Number of vases
Daisies used
Lilies used
Price of one vase
15
3
4
$11.00
5
9
12
$33.00
3
15
20
$55.00
© 2003 CompassLearning, Inc.
Activity 67115