Orchestra An orchestra conductor divides 48 violinists, 24 violists, and 36 cellists into ensembles....
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Orchestra An orchestra conductor divides 48 violinists, 24 violists, and 36 cellists into ensembles. Each ensemble has the same number of each instrument. What is the greatest number of ensembles that can be formed? How many violinists, violists, and cellists will be in each ensemble? In the Real World Greatest Common Factor 4 2 .
Transcript of Orchestra An orchestra conductor divides 48 violinists, 24 violists, and 36 cellists into ensembles....
Orchestra An orchestra conductor divides 48 violinists, 24 violists, and 36 cellists into ensembles. Each ensemble has the same number of each instrument. What is the greatest number of ensembles that can be formed? How many violinists, violists, and cellists will be in each ensemble?
In the Real World
Greatest Common Factor4 2.
Orchestra An orchestra conductor divides 48 violinists, 24 violists, and 36 cellists into ensembles. Each ensemble has the same number of each instrument. What is the greatest number of ensembles that can be formed? How many violinists, violists, and cellists will be in each ensemble?
In the Real World
One way to find the greatest common factor of two or more numbers is to make a list of all the factors of each number and identify the greatest number that is on every list.
Greatest Common Factor4 2.
A whole number that is a factor of two or more nonzero whole numbers is called a common factor.
The greatest of the common factors is called the greatest common factor (GCF).
In the previous slide, the greatest number of ensembles that can be formed is given by the greatest common factor of 48, 24, and 36.
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
The common factors are 1, 2, 3,
4, 6, and 12.
ANSWER
So, the greatest number of ensembles that can be formed is 12. Then each ensemble will have 4 violinists, 2 violists, and 3 cellists.
Making a List to Find the GCFEXAMPLE 1
Greatest Common Factor4 2.
The greatest common factor of 48, 24, and 36 is 12.
1, 2, 3, 4, 6,
1, 2, 3, 4, 6,
1, 2, 3, 4, 6, 12
12
12
The GCF is 12.
Greatest Common Factor4 2.
Using Prime Factorization
Another way to find the greatest common factor of two or more numbers is to use the prime factorization of each number.
The product of the common prime factors is the greatest common factor.
Find the greatest common factor of 180 and 126 using prime factorization.
Begin by writing the prime factorization of each number.
180
10 18
2 5 2 92 5 2 3 3
126
2 63
2 3 21
2 3 3 7
180 = 2 2 3 3 5
126 = 2 3 3 7
Greatest Common Factor4 2.
Using Prime Factorization to Find the GCFEXAMPLE 2
Find the greatest common factor of 180 and 126 using prime factorization.
Begin by writing the prime factorization of each number.
180
10 18
2 5 2 92 5 2 3 3
126
2 63
2 3 21
2 3 3 7
180 = 2 2 3 3 5
126 = 2 3 3 7
ANSWER
The common prime factors of 180 and 126 are 2, 3, and 3.
Greatest Common Factor4 2.
Using Prime Factorization to Find the GCFEXAMPLE 2
So, the greatest common factor is 2 3 2 = 18.
Relatively Prime Two or more numbers are relatively prime if their greatest common factor is 1.
Greatest Common Factor4 2.
Relatively Prime Two or more numbers are relatively prime if their greatest common factor is 1.
Tell whether the numbers are relatively prime.
28, 45 Factors of 28: 1, 2, 4, 7, 14, 28Factors of 45: 1, 3, 5, 9, 15, 45
The GCF is 1.
ANSWER
Because the GCF is 1, 28 and 45 are relatively prime.
15, 51 Factors of 15: 1, 3, 5, 15Factors of 51: 1, 3, 17, 51 The GCF is 3.
ANSWER
Because the GCF is 3, 15 and 51 are not relatively prime.
Greatest Common Factor4 2.
Identifying Relatively Prime NumbersEXAMPLE 3
