1) Greatest common factors (GCF) and least common multiples (LCM) are important concepts in mathematics. The GCF is the largest integer that divides two or more numbers, while the LCM is the smallest integer that is a multiple of two or more numbers. 2) There are several methods to find the GCF and LCM, including listing factors, using a factor tree, and continuous division. These methods involve breaking numbers down into their prime factors to identify common factors or multiples. 3) For the example of finding the GCF and LCM of 36 and 48, the GCF is 12 and the LCM is 144 using either the factor tree or continuous division methods.

1. Greatest Common Factors
and
Least Common Multiples

2. Greatest Common Factors
(GCF)

3. Factors of a Number
A number may be made by
multiplying two or more
other numbers together.

4. Factors of a Number
Factors of 24
1 X 24
2 X 12
3 X 8
4 X 6
{1, 2, 3, 4, 6, 8, 12, 24}
A list of its factors
Multiplication
combinations to
reach 24

5. Greatest Common Factor (GCF)
Greatest common factor
is the greatest or
biggest whole number
that is a factor of two or
more given whole
numbers.
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6. To find the Greatest Common Factor of a
number you can using PRIME
FACTORIZATION:
 Listing Method
 Factor Tree
 Continuous Division

7. What is the greatest common
factor of 36 and 48?

8. LISTING
METHOD

9. LISTING METHOD
is a process in which you
write down all the factors of
the given numbers.

10. What is the greatest common
factor of 36 and 48?
Factors of 36:
Factors of 48:
{1, 2, 3, 4, 6, 9, 12, 18, 36}
36
1 2 18
3 12
4 9
6
{1, 2, 3, 4, 6, 8,12, 16, 24, 48}

11. Factors of 36:
Factors of 48:
{1, 2, 3, 4, 6, 9, 12, 18, 36}
{1, 2, 3, 4, 6, 8,12, 16, 24, 48}
What is the greatest common
factor of 36 and 48?
Which factors are common to both
numbers?
12
12
6
6
4
4
3
3
2
2
1
1

12. Factors of 36:
Factors of 48:
{1, 2, 3, 4, 6, 9, 12, 18, 36}
{1, 2, 3, 4, 6, 8,12, 16, 24, 48}
What is the greatest common
factor of 36 and 48?
12
12
6
6
4
4
3
3
2
2
1
1
Which is the greatest common
factor?

13. FACTOR TREE
METHOD

14. FACTOR TREE METHOD
is a diagrams that show the
prime factors of numbers in a
branching manner.

15. - - Factor Tree Method - -
2 x 18
2 x 9
3 x 3
36 48

16. - - Factor Tree Method - -
2 x 18
2 x 9
3 x 3
48
2 x 24
2 x 12
2 x 6
36
2 x 3

17. - - Factor Tree Method - -
36 = 2 x 2 x 3 x 3
48 = 2 x 2 x 2 x 2 x 3
2 x 18
2 x 9
3 x 3
48
2 x 24
2 x 12
2 x 6
36
2 x 3

18. - - Factor Tree Method - -
36 = 2 x 2 x 3 x 3
48 = 2 x 2 x 2 x 2 x 3
Now that the numbers are written as a
product of their prime factors. Identify the
common prime factors then multiply it all the
product is the GCF.

19. - - Factor Tree Method - -
36 = 2 x 2 x 3 x 3
48 = 2 x 2 x 2 x 2 x 3
GCF= 2 x 2 x 3
GCF= 12

20. Continuous
Division

21. CONTINUOUS DIVISION
is the process of dividing a
number by possible prime factors
until there is no other number
besides 1 that is a factor of both
numbers.

22. CONTINUOUS DIVISION
36 48
Think of any number
that divides evenly
into both 36 and 48
(a factor of both)
For this example, I’ll use 2.
2
18 24

23. 36 48
2
18 24
CONTINUOUS DIVISION
Think of any number
that divides evenly
into both 18 and 24
(a factor of both)
For this example, I’ll use 3.
3
Divide both numbers by 3.
6 8
2

24. 36 48
2
18 24
CONTINUOUS DIVISION
Continue this process until
there is no other number
besides 1 that is a factor of
both numbers.
3
3 and 4 have no common factor
except
1... so you can stop dividing.
6 8
2
2
3 4

25. CONTINUOUS DIVISION
18 24
3
6 8
2
2
3 4
36 48
The product of the
numbers along the
LEFT SIDE is the
GCF of 36 and 48.
GCF= 2 x 3 x 2
GCF = 12

26. Least Common Multiples
(LCM)

27. 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.
For example:
7 x 1 = 7
7 x 2 = 14
7 x 3 = 21
7 x 4 = 28
So, the multiples of 7 are
7, 14, 21, 28, and so on.

28. 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, …
Multiples of a Number

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

30. To find the LEAST COMMON
MULTIPLE of a number you can using
PRIME FACTORIZATION:
 Listing Method
 Factor Tree
 Continuous Division

31. LISTING
METHOD

32. LISTING METHOD
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.

33. 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

34. What is the Least Common
Multiple of 36 and 48?

35. Least Common Multiple (LCM)
Find the least common multiple of 36
and 48.
Solution:
List the multiples of 36 and stop when
you find a multiple of 48.
36 = 36,72, 108, 144, 180, …
48 = 48, 96, 144, …
LCM is
144

36. Prime Factorization to find LCM
With the Factor Tree and Continuous Division
methods, you can find both the GCF and LCM at
the same time!
Let’s use a previous example…

37. Find the least
common multiple
of 36 and 48.

38. FACTOR TREE
METHOD

39. - - Factor Tree Method - -
2 x 18
2 x 9
3 x 3
36 48

40. - - Factor Tree Method - -
2 x 18
2 x 9
3 x 3
48
2 x 24
2 x 12
2 x 6
36
2 x 3

41. - - Factor Tree Method - -
36 = 2 x 2 x 3 x 3
48 = 2 x 2 x 2 x 2 x 3
2 x 18
2 x 9
3 x 3
48
2 x 24
2 x 12
2 x 6
36
2 x 3

42. - - Factor Tree Method - -
36 = 2 x 2 x 3 x 3
48 = 2 x 2 x 2 x 2 x 3
Now that the numbers are written as a
product of their prime factors. Multiply all the
prime factors. Take note, common factor should
be written once only.

43. - - Factor Tree Method - -
36 = 2 x 2 x 3 x 3
48 = 2 x 2 x 2 x 2 x 3
LCM = 2 x 2 x 3 x 2 x 2 x 3

44. - - Factor Tree Method - -
LCM = 2 x 2 x 3 x 2 x 2 x 3
LCM = 4 x 3 x 2 x 2 x 3
LCM = 12 x 2 x 2 x 3
LCM = 24 x 2 x 3
LCM = 48 x 3
LCM = 144

45. Continuous
Division

46. CONTINUOUS DIVISION
36 48
Think of any number
that divides evenly
into both 36 and 48
(a factor of both)
For this example, I’ll use 2.
2
18 24

47. 36 48
2
18 24
CONTINUOUS DIVISION
Think of any number
that divides evenly
into both 18 and 24
(a factor of both)
For this example, I’ll use 3.
3
Divide both numbers by 3.
6 8
2

48. 36 48
2
18 24
CONTINUOUS DIVISION
Continue this process until
there is no other number
besides 1 that is a factor of
both numbers.
3
3 and 4 have no common factor
except
1... so you can stop dividing.
6 8
2
2
3 4

49. CONTINUOUS DIVISION
18 24
3
6 8
2
2
3 4
36 48
The product of the
numbers outside the
line is the Least
Common Multiple
(LCM).
LCM = 2 x 3 x 2 x 3 x 4
LCM = 144

50. MATHEMATICS
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