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 is the greatest whole number that is a factor of two or more given whole numbers.
Example
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4. What is the greatest common factor of 36 and 48?
Factors of 36:
{1, 2, 3, 4, 6, 9, 12, 18, 36} 1 2 3 4 6 9 12 18 36
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} 1 2 3 4 6 12
Factors of 48:
{1, 2, 3, 4, 6, 8,12, 16, 24, 48} 1 2 3 4 6 12
Which is the greatest common factor?
Which factors are common to both numbers?

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 that’s easier?
Duh!

7. Birthday Cake Method to find GCF
Think of any number that divides evenly into both 36 and 48
(a factor of both)
Divide both numbers by 2.
For this example, I’ll use 2. 2

8. Birthday Cake Method to find GCF
Think of any number that divides evenly into both 18 and 24
(a factor of both)
Divide both numbers by 3. 6 8
For this example, I’ll use 3. 3 2

9. Birthday 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 2 6 8 3 2

10. 2 x 3 x 2 = 12 Birthday Cake Method to find GCF 3 4 2 6 8 3 18 24 2
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 2 6 8 3 2

11. 8 x 2 = 16 Birthday Cake Method to find GCF
Observe another example. Find the
GCF of 64 and 80.
8 x 2 = 16 4
no common
factor except 1 5 8 2

12. 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.

13. Multiples of a Number multiples of 2 are 2, 4, 6, 8, …
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, …

14. Common Multiples
Multiples that are common to two or more numbers are said to be common multiples. For example:
Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, …
Multiples of 3 are 3, 6, 9, 12, 15, 18, …
So, common multiples of 2 and 3 are 6, 12, 18, … 6 12 18 12 18 6

15. Example 2:
Find the common multiples of 4 and 6.
Solution:
Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, … Multiples of 6 are 6, 12, 18, 24, 30, 36, …
So, the common multiples of 4 and 6 are 12, 24, 36, … 12 24 36 12 24 36

16. Least Common Multiple (LCM)
The smallest common multiple of two or more numbers is called the least common multiple (LCM).

17. Least Common Multiple (LCM)
To find the least common multiple (LCM) of two or more numbers, list the multiples of the larger number and stop when you find a multiple of the other number.  This is the LCM.

18. Least Common Multiple (LCM)
Find the least common multiple of 6 and 9.
Solution:
List the multiples of 9 and stop when you find a multiple of 6.
Multiples of 9 are 9, 18, …
Multiples of 6 are 6, 12, 18, …
                      
LCM is 18

19. Birthday Cake Method to find LCM
With the birthday cake method, you can find both the GCF and LCM at the same time!
Let’s use a previous example…

20. Birthday Cake Method to find LCM
Find the least common multiple of 36 and 48.

21. Birthday Cake Method to find LCM
Find the product of the
numbers going up and on
top of the cake.
Let’s make the birthday cake! 3 4
= 144
144 is the LCM
of 36 and 48 x x 2 6 8 x 3 x 2

22. Birthday 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 don’t. So their LCM is their product… ?
16 9 X
= 144

23. Birthday Cake Method to find LCM
Another example: Find the LCM of 5, 6 and 8.
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.
Finding the answer by listing multiples of the largest number (8)
would eventually lead you to the answer of 120.

24. Birthday Cake Method to find LCM
The birthday cake method can be used to find the LCM of 5, 6 and 8. Watch! 5
3 4 x x
= 120 x 2