Factors and Multiples

1
Factors and Multiples
2
Factors of a Number

• A number may be made by multiplying two or more
  other numbers together. The numbers that are
  multiplied together are called factors of the
  final number.

Factors of 24
1 X 24 2 X 12 3 X 8 4 X 6
Multiplication combinations to reach 24
1, 2, 3, 4, 6, 8, 12, 24 A list of its factors
3
Greatest Common Factor (GCF)

• Greatest common factor or GCF is the greatest
  whole number that is a factor of two or more
  given whole numbers.

Example
next page
4
What is the greatest common factor of 36 and 48?

• Factors of 36

1, 2, 3, 4, 6, 9, 12, 18, 36
36
1
2
18
3
12
4
9
6
Factors of 48
1, 2, 3, 4, 6, 8,12, 16, 24, 48
5
What is the greatest common factor of 36 and 48?
Factors of 36
1, 2, 3, 4, 6, 9, 12, 18, 36
12
6
4
3
2
1
Factors of 48
1, 2, 3, 4, 6, 8,12, 16, 24, 48
12
6
4
3
2
1
Which factors are common to both numbers?
Which is the greatest common factor?
6
Greatest Common Factor (GCF)

• That was too much work. What happened if we
  forgot to include the 12 in one of the factor
  list?

Want to learn a different way thats easier?
Duh!
7
Cake Method to find GCF
Think of any prime number that divides evenly
into both 36 and 48 (a factor of both)
Divide both numbers by 2.
For this example, Ill use 2.
18 24
36 48
2
8
Cake Method to find GCF
Think of any prime number that divides evenly
into both 18 and 24 (a factor of both)
Divide both numbers by 3.
6
8
For this example, Ill use 3.
3
18 24
36 48
2
9
Cake Method to find GCF
Continue this process until there is no other
number besides 1 that is a factor of both numbers.
3 and 4 have no common factor except 1... so you
can stop dividing.
3
4
6
8
2
3
18 24
36 48
2
10
Cake Method to find GCF
The product of the numbers along the side of the
cake is the GCF of 36 and 48.
2 x 3 x 2 12
3
4

6
8
2
3
18 24
36 48
2
11
Cake Method to find GCF
Observe another example. Find the GCF of 64 and
80.
8 x 2 16
4
5
no common factor except 1
8
32 40
64 80
2
12
More Practice

• Use the cake method to find the GCF of 14 and 35.
• Use the cake method to find the GCF of 42 and 72.
• Use the cake method to find the GCF of 12 and 25.

13
More, More Practice!! ?

• Jane and her friends are going on a hiking trip
  on the Appalachian Trail. She wants to take
  snack bags. She has 36 bananas and 24 bags of
  peanuts. What is the greatest number of snack
  bags she can make if each bag has the exact same
  number of bananas and bags of peanuts? She
  doesnt want any bananas or bags of peanuts left
  over. Explain your answer.

14
More, More, More Practice ?

• Suppose that Janes dog ate 6 bananas. Now what
  is the greatest number of snack bags she can make
  with no leftovers?

15
Still More Practice

• Ashley has 27 violet marbles, 54 blue marbles,
  and 72 white marbles. She wants to divide the
  marbles into groups so that each group has the
  same number of each color marble. What is
  greatest number of groups that Ashley can make?
  Explain.

16
Multiples of a Number
The products of a number with the natural numbers
1, 2, 3, 4, 5, ... are called the multiples of
the number.
7 x 1 7 7 x 2 14 7 x 3 21 7 x 4 28
For example
So, the multiples of 7 are 7, 14, 21, 28, and so
on.
17
Multiples of a Number

• Further examples
• multiples of 2 are 2, 4, 6, 8,
• multiples of 3 are 3, 6, 9, 12,
• multiples of 4 are 4, 8, 12, 16,

18
Least Common Multiple (LCM)

• The smallest common multiple of two or more
  numbers is called the least common multiple
  (LCM).

19
Cake Method to find LCM
With the cake method, you can find both the GCF
and LCM at the same time!
Lets use a previous example
20
Cake Method to find LCM
Find the least common multiple of 36 and 48.
21
Cake Method to find LCM
Find the product of the numbers going up and on
top of the cake.
Lets make the cake!
144 144 is the LCM of 36 and 48
3
4
x
x
6
8
2
x
3
18 24
x
36 48
2
22
Cake Method to find LCM
Another example Find the LCM of 16 and 9.
Do 16 and 9 have any common factors besides 1?
No, they dont. So their LCM is their product
16 9
?
X
144
23
Cake Method to find LCM
Another example Find the LCM of 5, 6 and 8.
Finding the answer by listing multiples of the
largest number (8) would eventually lead you to
the answer of 120.
Multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64,
72, 80, 88, 96, 104, 112, 120, Stop at 120 as
it is a multiple of both 5 and 6.
24
Cake Method to find LCM
The cake method can be used to find the LCM of 5,
6 and 8. Watch!
3 4
5
x
x
120
x
5 6 8
2