Transcript LCM

Least Common Multiples
And Greatest Common
Factor
1
Lesson
3.4.5
Least Common Multiples
California Standards:
What it means for you:
Number Sense 2.1
Solve problems involving addition,
subtraction, multiplication, and
division of positive fractions and
explain why a particular operation
was used for a given situation.
Number Sense 2.4
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 for a fraction).
You’ll learn how to find the
lowest number that two
numbers both divide into
exactly. You’ll also see how
this can help you with adding
and subtracting fractions.
Key words:
• least common multiple
• greatest common factor
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Lesson
3.4.5
Least Common Multiples
Today
weCommon
are going
about
The
Least
ulteto
Is learn
Less Than
theLowest
Others
Common Multiples and Greatest Common
Factor. First let’s examine the parts of a
multiplication equation.
MULTIPLE
3
Lesson
3.4.5
Least Common Multiples
Definition
Least Common Multiple (LCM): The lowest
multiple of two or more numbers.
Example: What is the
LCM of 10 and 30?
10 x 1 = 10
10 x 2 = 20
10 x 3 = 30
10 x 4 = 40
10 x 5 = 50
10 x 6 = 60
30 x 1 = 30
30 x 2 = 60
30 x 3 = 90
So the least common multiple of 10 and 30 is 30.
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Lesson
3.4.5
Least Common Multiples
When
doCommon
you useulte
LCM?
The
Least
Is Less Than the Others
2 + 1
10
30
When you need to add fractions with
different denominators!
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Lesson
3.4.5
Least Common Multiples
Definition
Greatest Common Factor (GCF): The largest
factor shared by two or more numbers.
Example: What is the
GCF of 20 and 30?
1 x 20
2 x 10
4x5
1 x 30
2 x 15
3 x 10
5x6
The factors of 20 are 1, 2, 4, 5, 10, and 20
The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30
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Lesson
Least Common Multiples
3.4.5
When
doCommon
you useulte
GCF?
The
Least
Is Less Than the Others
20
30
When you need to SIMPLIFY fractions!
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Lesson
3.4.5
Least Common Multiples
You
can find the
The LeastThan
the Lowest
Others Common Multiple
(LCM) and the Greatest Common Factor (GCF)
with a factor tree. Let’s try some!
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Lesson
3.4.5
Example
Least Common Multiples
1
Find the LCM and GCF of 20 and 12.
Solution
So the least common multiple of 20 and 12 is 60 .
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Solution follows…
Lesson
3.4.5
Example
Least Common Multiples
2
Find the LCM and GCF of 16 and 28.
Solution
So the least common multiple of 16 and 28 is 112.
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Solution follows…
Lesson
3.4.5
Example
Least Common Multiples
3
Find the LCM and GCF of 12 and 44.
.
Solution
So the least common multiple of 12 and 44 is:
12 × 44 ÷ 4 = 132
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Solution follows…
Lesson
3.4.5
Least Common Multiples
Guided Practice
Find the least common multiple and greatest common factor.
1. 4, 40
2. 15, 50
3. 12, 9
4. 10, 25
12
Solution follows…
Lesson
3.4.5
Least Common Multiples
Independent Practice
Find the LCM and GCF.
7. 30, 45
8. 4, 12
9. 14, 21
10. 28, 30
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Solution follows…
LCM and GCF: Fractions
with Different
Denominators
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Lesson
LCM and GCF
3.4.5
Example
1
1
1
What is 3 + 5 ?
Solution:
1
1
5
3
5+3
8
+
=
=
=
+
3
5
15 15
15
15
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Lesson
LCM and GCF
3.4.5
Example
2
3
5
What is 40 - 16 ?
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Lesson
3.4.5
Example
LCM and GCF
3
What is 2 ½ + 3 ¼ =
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Lesson
3.4.5
Least Common Multiples
Guided Practice
Use factor trees to find the GCF and LCM. Solve the
equations once they have the same denominator.
1.
13
9
=
+
10
15
3.
1
5
=
+
6
10
39 18 57 19
+
=
=
30 30 30 10
5
15
20
2
+
=
=
30 30
30
3
2.
4
5= 8 +
+
9
6 18
4.
5
1 = 20 – 3 = 17
–
6
8 24 24 24
15 23
=
18 18
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Solution follows…
Lesson
3.4.5
Least Common Multiples
Guided Practice
Solve the following problems.
5.
3
2
=
+
15
20
6.
4
2
=
–
6
8
12
6
18
3
+
=
=
60 60 60 10
16
6
10
5
–
=
=
24 24 24 12
13
5
=
+
42
14
13 15 28 2
+
=
=
42 42 42 3
**TICKET
PROBLEM
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Solution follows…
Lesson
3.4.5
Least Common Multiples
Independent Practice
Find the LCM and GCF for problems 7-15 by making
factor trees. Then add or subtract the fractions.
2 1
–
3 4
3 1
–
4 2
5
12
8.
10. 2 + 15
10 25
4
5
11. 8 – 5
9 6
8
18
+
11 44
25
22
14.
7.
13.
13 12
+
15 18
1 2
+
9 7
1
4
9.
1
18
12. 13 – 3
12 4
23
15
15.
9
9
–
10 14
25
63
1
3
9
35
20
Solution follows…
Lesson
3.4.5
Least Common Multiples
Independent Practice
Work out the sums and differences in Exercises 26–31.
Give your answers in their simplest form.
26.
7
15
+
75 35
29. 10 + 6
49 21
274
525
27. 4 – 1
27 45
24
49
30.
7 1
+
9 30
17
135
28. 1 + 15
24 56
13
42
73
90
31. 31 – 10
42 56
47
84
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Solution follows…
Lesson
3.4.5
Least Common Multiples
Round Up
Pausing for a second to find the least common
multiple before trying to add or subtract two
fractions with different denominators is a good way
to avoid having to deal with really big numbers.
Smaller numbers also make simplifying at the
end easier.
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