Transcript GCF and LCM notes

Finding GCF’s and LCM’s
and some Applications
Finding the Greatest Common Factor
of Two Numbers
We are looking for a factor. The factor
must be common to both numbers. We
need to pick the greatest of such
common factors.
The GCF of 36 and 90
Method 1
1) List the factors of each number.
36: 1
2
3
4
36 18
24
9
2
3
5
90 45
30
90: 1
6
6
9
18 15 10
2) Circle the common factors.
3) The greatest of these will be your Greatest Common Factor:
18
The GCF of 36 and 90
Method 2
1) Prime factor each number.
36 = 2 ● 2 ● 3 ● 3
90 = 2 ● 3 ● 3 ● 5
2) Circle each pair of common prime factors.
3) The product of these common prime factors will be
the Greatest Common Factor:
2 ● 3 ● 3 = 18
Finding the Least Common Multiple
of Two Numbers
We are looking for a multiple. The multiple
must be common to both numbers. We
need to pick the least of such
common multiples.
The LCM of 12 and 15
Method 1
1) List the first few multiples of each number.
12: 12
24
15: 15 30
36 48 60 72 84 90 108 120
45
60
75
90
105
120 135
2) Circle the common multiples.
3) The least of these will be your Least Common Multiple:
60
The LCM of 12 and 15.
Method 2
1) Prime factor each number.
12 = 2 ● 2 ● 3
15 = 5 ● 3
2) Circle each pair of common prime factors.
3) Circle each remaining prime factor.
4) Multiply together one factor from each circle to get the
Least Common Multiple :
3 ● 2 ● 2 ● 5 = 60
Note that the common factor, 3, was only used once.
Method 3: Find both GCF and LCM at Once.
The GCF and LCM of 72 and 90
1) Make the following table.
9
72
2
8
4
90
10
5
2) Divide each number by a common factor.
3) Divide the new numbers by a common factor.
Repeat this process until there is no longer a common factor.
The product of the
factors on the left
is the GCF:
9 ● 2 = 18
The product of the
factors on the left AND
bottom is the LCM:
9 ● 2 ● 4 ● 5 = 360
Method 3: Find both GCF and LCM at Once.
One more example: The GCF and LCM of 96 and 144
1) Make the following table.
Note that you can pick
any common factor to
start and any
remaining common
factor for each step.
Try starting by dividing
by 3 to see that this is
so.
2
96
144
6
4
48
8
2
72
12
3
2) Divide each number by a common factor.
3) Divide the new numbers by a common factor.
4) Repeat this process until there is no longer a common factor.
The product of the
factors on the left
is the GCF:
2 ● 6 ● 4 = 48
The product of the
factors on the left AND
bottom is the LCM:
2 ● 6 ● 4 ● 2 ● 3 = 288
Make sure that you
are able to use
each of the methods
described in this
lecture.
We will now do a
couple of
applications
Work on this problem:
Juan, Sean and Jane are night guards at
an industrial complex. Each starts work at
the central gate at 12 midnight. Each
guard spends the night repeating a round
which starts and ends at the gate. Juan’s
round takes 30 minutes; Sean’s round
takes 40 minutes; and Jane’s round takes
80 minutes. If they all head out from the
gate at midnight, what is the next time
that they will all be at the gate.
Juan, will return at 12:30, 1:00, 1:30 and so forth.
Sean, will return at 12:40, 1:20, 2:00 and so forth.
Jane, will return at 1:20, 2:40, 4:20 and so forth.
Working with times can be awkward. It is best to work with minutes.
Juan, will return after 30 minutes, 60 minutes, 90
minutes, and so forth.
Sean, will return after 40 minutes, 80 minutes, 120
minutes, and so forth.
Jane, will return after 80 minutes, 160 minutes, 240
minutes, and so forth.
You should recognize this as an application of the
Least Common Multiple.
Juan: 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, …
Sean: 40, 80, 120, 160, 200, 240, 280, 320, …
Jane: 80, 160, 240, 320, …
After 240 minutes they are all at the gate.
You can also model the rounds this way.
Juan
Sean
Jane
30
30
30
30
30
30
30
30
minutes minutes minutes minutes minutes minutes minutes minutes
40
minutes
40
minutes
80
minutes
40
minutes
40
minutes
80
minutes
40
minutes
40
minutes
80
minutes
After four rounds for Juan and three rounds for Sean, they are
both back at the gate. Every time Jane comes back to the gate,
Sean is there. It is only in 240 minutes, after Juan has made 8
rounds, Sean has made 6 rounds and Jane has made 3 rounds,
that all three meet at the gate.
What have we forgotten?
Juan, Sean and Jane are night guards at an industrial complex. Each
starts work at the central gate at 12 midnight. Each guard spends the
night repeating a round which starts and ends at the gate. Juan’s
round takes 30 minutes; Sean’s round takes 40 minutes; and Jane’s
round takes 80 minutes. If they all head out from the gate at midnight,
what is the next time that they will all be at the gate.
We know that the guards meet at the gate again after 240
minutes, however the problem asks for a time.
240 minutes divided by the 60 minutes in an hour give us 4 hours.
4 hours after 12 midnight is 4 a.m.
The guards meet at the gate again at 4 a.m.
Now work on this problem:
You neighbor is putting down a floor with
rectangular pieces of plywood. Each
piece of plywood is 6 feet by 8 feet. If the
floor is square, what is the least possible
number of plywood pieces used? Draw a
diagram of the situation and solve.
Start with one 6 x 8 board and add boards to the right and below
until you have a square. You will need to click to add boards.
8’
8’
6’
16’
24’
8’
8’
6’
12’
18’
6’
24’
6’
6’
We have our square floor. It is 24 feet by 24 feet. It uses 4 x 3 = 12 boards.
The area of the floor is 24 x 24 = 576 square feet.
Reread the problem to remember what it asked us to find.
You neighbor is putting down a floor with rectangular
pieces of plywood. Each piece of plywood is 6 feet by 8
feet. If the floor is square, what is the least possible
number of plywood pieces used? Draw a diagram of
the situation and solve.
We need to find the minimum number of boards that will
make a square floor: 12 boards are needed to make a
square floor.
Make sure that you work the
problems in the exercises!
Right click and
select End Show.
Then Close to
return.