Transcript LCM - TeacherWeb

Least Common Multiples
Looking “Forward” with the
Values
What is our objective?
• Today we will work with least common
multiples (LCM).
• We will find the LCM for a set of numbers.
• We will also use the LCM of a set of numbers
to solve word problems.
• Please remember……a common multiple is
needed when working with unlike fractions!
Vocabulary
Common multiples:
Common multiples are multiples that are
shared by a set of numbers. (notes)
Least common multiple:
The smallest, shared multiple…..
Why do we use the LCM?
• We use the LCM to
create common
denominators.
Common denominators
are needed to add and
subtract fractions.
• We also use common
denominators to solve
word problems
involving different
values that must be
distributed evenly.
• These word problems
often involve unlike
fractions!
First things first….finding the LCM
The first method is the one You have probably
used before.
Step 1: List multiples of each number.
12: 12, 24, 36, 48, 60, 72, 84…
36: 36, 72, 108 ...
Step 2: Identify the common multiples
Step 3: Find the least multiple shared
36 is the LCM
First things first….finding the LCM
Step 1: List multiples of each number.
5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55 ….
4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48 . . .
Step 2: Identify the common multiples
Step 3: Find the least multiple shared
20 is the LCM
In our last example, 20 was the LCM. When the
numbers you are working with do NOT share
any common factors, the product of the numbers
is the LCM. Here are a few examples of this.
You may want to put these in your notes.
3 and 5 share no factors: LCM = 15
7 and 5 share no factors: LCM = 35
8 and 9 share no factors: LCM = 72
10 and 9 share no factors: LCM = 90
Good to Know….
If you are looking for the LCM, and one number is
a factor of another, just use the large number.
For example, if you are looking for the LCM of 4 and 16,
since 4 is found “inside” 16, 16 is the LCM.
12
24 Since 12 is a factor of 24, just use the 24 and 18 !!
18
12
6 Since 6 is found “inside” 12, just use the 12 and 30
30
The second method is helpful when the values
you are working with share factors.
This method uses the prime factorization
method.
Let’s look at 12 and 20. Factors are sharedHere’s
another
by these numbers.
way!!
2 x 3 x 2 x 5 = 60 LCM
12: ( 6 x 2) = 2 x 3 x 2
20: ( 4 x 5) = 2 x
2 x 5
12 4
20
We now have a complete list of all factors, but
we will not duplicate the shared factors.
3
5
4 x 3 x 5 = 60
Because the second method is new, let’s try
another example. Find the LCM for 18, 24, and 30.
2 x 3 x 3 x2 2x 5=
18: ( 6 x 3) = 2 x 3 x 3
24: ( 6 x 4) = 2 x 3
2 x 2
30: ( 6 x 5) = 2 x 3
5
Multiply the “value” of each swim lane. This keeps you
from duplicating any shared values.
360 is the LCM.
4 x 3 x 5 x 6 = 360
18
24 6
30
3
4
5
Let’s try another example. Three methods will
be shown. Find the LCM for 12, 20, and 8
2
8: ( 4 x 2) = 2
20: ( 5 x 4) = 2
12: ( 4 x 3) = 2
x 2 x 2 x 5 x 3 = 120
x 2 x 2
x 2
5
x 2
3
8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104,
112, 120
20: 20, 40, 60, 80, 100, 120
12: 12,24,36,48,60,72,84,96,108,120
2
8
12
20
X
3
4
4 x2 x 3 x 5 = 120
5
Be careful with this method. If you do not start with the GCF, your multiple
will NOT be the LCM………
AND if the “pink” numbers also
share a factor, they you will
get a number that is too big!!
6
24
16
20
4
4
5
2x3
2x2
Partner Practice
For each number set, find the LCM. Try using each method.
3 X 2 x 7
6 ( 3 x 2) 3 x 2
21 (3 x 7) 3 x 7
6, 12, 18, 24, 30, 36, 42
21, 42
2x2x2x5
8 ( 4 x 2) 2 x 2 x 2
x 5
10 (5 x 2) 2
LCM 42
8, 16, 24, 32, 40
10, 20, 30, 40
LCM 40
7 These numbers share NO factors:
LCM 56
7 x 8 = 56
8
How is this used?????
Consumer Application: Group Discussion
English muffins come in packs of 8, and eggs in cartons of 12.
If there are 24 students, what is the least number of packs and
cartons needed so that each student has a muffin sandwich
with one egg and there are no muffins left over?
Think of muffins in groups of 8. Think of eggs in groups of
12. We are looking for the smallest multiple they both
share.
They have given us the LCM of 24!!! We are being asked,
“What multiple of 8 is 24?” 8n = 24?
So 3 packs of English muffins are
needed.
They have given us the LCM of 24!!!
We are being asked,“What multiple of 12 is 24?”
12n = 24? So 2 cartons of eggs are needed.
There are 24 English muffins and 24 eggs.
Group Discussion
Here is another example question using LCM.
Dog cookies come in packages of 6, and bones in bags of 9. If
there are 18 dogs, what is the least number of packages and
bags needed so that each dog has a treat box with one bone
and one cookie and there are no bones or cookies left over?
Think of cookies in groups of 6. Think of bones in groups of 9.
You are looking for the first multiple they share.
They have given us the LCM of 18!!! How many of each item do we
need?
There are 18 dog cookies and 18 bones.
So 3 packages of dog cookies and 2 bags of bones are needed.
Your Turn! Let’s Apply the Skill
Linda is sending out invitations. If envelopes come in
boxes of 25, and stamps come in packs of 10, what is
the least number of stamps and envelopes she
can buy to get one stamp for each envelope?
10:
10, 20, 30, 40, 50
Listing the multiples to find the
works50
best here!!
25:GCF25,
Linda needs 50 stamps and
envelopes.
How many boxes of envelopes???? 2
How many packs of stamps???? 5
Film Title
Length
Introduction to the Museum
2 minutes
Profiles of Artists
30 minutes
Art and Architecture
45 minutes
Films play continuously at the museum. If the three films shown in the
table above begin to play at the same time at 8:00 a.m., what time will it
be before they begin playing together again? (Both methods are shown)
We know the 2 is found inside the 30. All you need to use are the 30 and 45.
30: 30, 60, 90
45: 45, 90
3 X 2x 5 x 3
30: (3 x 10) 3 x 2 x 5
45: (9 x 5) 3
5
3
LCM = 90 minutes
The films will start again together
in 90 minutes (1 ½ hours), or 9:30 a.m.
Did we meet the objectives?
• Did we find the LCM for sets of numbers?
• Did we use the LCM to answer word
problems?