Transcript Find the greatest common divisor

3-2 Greatest Common Divisor
California
Standards
NS2.4 Determine the least common
multiple and the greatest common divisor of
whole numbers; use them to solve problems
with fractions (e.g. to find a common
denominator to add two fractions or to find the
reduced form of a fraction).
Holt CA Course 1
3-2 Greatest Common Divisor
Vocabulary
greatest common divisor (GCD)
The greatest whole number that divides evenly
into two or more whole numbers (also called
GCF-Greatest Common Factor)
Holt CA Course 1
3-2 Greatest Common Divisor
The greatest common divisor (GCD) of two
or more whole numbers is the greatest whole
number that divides evenly into each number.
One way to find the GCD of two or more numbers
is to list all the factors of each number. The GCD
is the greatest factor that appears in all the lists.
Holt CA Course 1
3-2 Greatest Common Divisor
Example 1: Using a List to Find the GCD
Find the greatest common divisor (GCD) of 12,
36, and 54.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
List all of the factors of each number.
Circle the greatest factor that is in all the lists.
The GCD is 6.
Holt CA Course 1
3-2 Greatest Common Divisor
Example 2A: Using Prime Factorization to Find the
GCD
Find the greatest common divisor (GCD).
40, 56
40 = 2  2  2  5
56 = 2  2  2  7
222=8
The GCD is 8.
Holt CA Course 1
Write the prime factorization of
each number and circle the
common prime factors.
Multiply the common prime
factors.
3-2 Greatest Common Divisor
Example 2B: Using Prime Factorization to Find the GCD
Find the greatest common divisor (GCD).
252, 180, 96, 60
Write the prime factorization
of each number and circle
180 = 2  2  3  3  5 the common prime factors.
252 = 2  2  3  3  7
96 = 2  2  2  2  2  3
60 = 2  2  3  5
2  2  3 = 12
The GCD is 12.
Holt CA Course 1
Multiply the common prime
factors.
3-2 Greatest Common Divisor
Example 3: Problem Solving Application
You have 120 red beads, 100 white
beads, and 45 blue beads. You want to
use all the beads to make identical
bracelets that have red, white, and blue
beads on each. What is the greatest
number of matching bracelets you can
make?
Holt CA Course 1
3-2 Greatest Common Divisor
Additional Example 3 Continued
1
Understand the Problem
Rewrite the question as a statement.
• Find the greatest number of bracelets you can
make.
List the important information:
• There are 120 red beads, 100 white beads,
and 45 blue beads.
• Each bracelet must have the same
number of red, white, and blue beads.
The answer will be the GCD of 120, 100, and 45.
Holt CA Course 1
3-2 Greatest Common Divisor
Additional Example 3 Continued
2
Make a Plan
You can list the prime factors of 120, 100,
and 45 to find the GCD.
3
Solve
120 = 2  2  2  3  5
100 = 2  2  5  5
Circle the prime
factors that are
common to 120,
100, and 45 .
45 = 3  3  5
The GCD of 120, 100, and 45 is 5.
You can make 5 bracelets.
Holt CA Course 1
3-2 Greatest Common Divisor
Additional Example 3 Continued
4
Look Back
If you make 5 bracelets, each one will have
24 red beads, 20 white beads, and 9 blue
beads, with no beads left over.
Holt CA Course 1
3-2 Greatest Common Divisor
Check It Out! Example 1
Find the greatest common divisor of 14, 28,
and 63.
14: 1, 2, 7, 14
28: 1, 2, 4, 7, 14, 28
63: 1, 3, 7, 9, 21, 63
List all of the factors of each number.
Circle the greatest factor that is in all the lists.
The GCD is 7.
Holt CA Course 1
3-2 Greatest Common Divisor
Check It Out! Example 2
Find the greatest common divisor (GCD).
360, 250, 170, 40
360 = 2  2  2  3  3  5
250 = 2  5  5  5
170 = 2  5  17
40 = 2  2  2  5
2  5 = 10
The GCD is 10.
Holt CA Course 1
Write the prime
factorization
of each number and
circle the common
prime factors.
Multiply the common prime
factors.
3-2 Greatest Common Divisor
Check It Out! Example 3
Nathan has made fishing flies that he
plans to give away as gift sets. He has
24 wet flies and 18 dry flies. Using all of
the flies, how many sets can he make?
Holt CA Course 1
3-2 Greatest Common Divisor
Check It Out! Example 3 Continued
1
Understand the Problem
Rewrite the question as a statement.
• Find the greatest number of sets of flies
he can make.
List the important information:
• There are 24 wet flies and 18 dry flies.
• He must use all of the flies.
The answer will be the GCD of 24 and 18.
Holt CA Course 1
3-2 Greatest Common Divisor
Check It Out! Example 3 Continued
2
Make a Plan
You can list the prime factors of 24 and 18
to find the GCD.
3
Solve
24 = 2  2  2  3
18 = 2  3  3
Multiply the prime factors
23=6
that are common to both
24 and 18.
You can make 6 sets of flies.
Holt CA Course 1
3-2 Greatest Common Divisor
Check It Out! Example 3 Continued
4
Look Back
If you make 6 sets, each set will have
3 dry flies and 4 wet flies.
Holt CA Course 1