The document discusses prime factorization, which is the process of writing a composite number as a product of prime numbers. It defines natural numbers, multiples, factors, and prime numbers. A prime number is only divisible by 1 and itself, while composite numbers can be divided by other numbers. The document provides examples of finding all the factors of 12 and listing its multiples. It states that a complete factorization is when all factors are prime numbers, like writing 12 as 2 * 2 * 3.

1. Prime Factorization

2. Whole numbers 1, 2, 3, 4,.. are called natural numbers and
they are used to track whole items.
Prime Factorization

3. Whole numbers 1, 2, 3, 4,.. are called natural numbers and
they are used to track whole items.
If x and y are two natural numbers and x can be divided by y,
we say x is a multiple of y
Prime Factorization

4. Whole numbers 1, 2, 3, 4,.. are called natural numbers and
they are used to track whole items.
If x and y are two natural numbers and x can be divided by y,
we say x is a multiple of y or y is a factor of x.
Prime Factorization

5. Whole numbers 1, 2, 3, 4,.. are called natural numbers and
they are used to track whole items.
If x and y are two natural numbers and x can be divided by y,
we say x is a multiple of y or y is a factor of x. Hence 15 is a
multiple of 3 (or 5),
Prime Factorization

6. Whole numbers 1, 2, 3, 4,.. are called natural numbers and
they are used to track whole items.
If x and y are two natural numbers and x can be divided by y,
we say x is a multiple of y or y is a factor of x. Hence 15 is a
multiple of 3 (or 5), and 3 (or 5) is a factor of 15.
Prime Factorization

7. Whole numbers 1, 2, 3, 4,.. are called natural numbers and
they are used to track whole items.
If x and y are two natural numbers and x can be divided by y,
we say x is a multiple of y or y is a factor of x. Hence 15 is a
multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a
number x is always the multiple of itself and the number 1.
Prime Factorization

8. Whole numbers 1, 2, 3, 4,.. are called natural numbers and
they are used to track whole items.
If x and y are two natural numbers and x can be divided by y,
we say x is a multiple of y or y is a factor of x. Hence 15 is a
multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a
number x is always the multiple of itself and the number 1.
A prime number is a natural number that can only be divided
by 1 and itself.
Prime Factorization

9. Whole numbers 1, 2, 3, 4,.. are called natural numbers and
they are used to track whole items.
If x and y are two natural numbers and x can be divided by y,
we say x is a multiple of y or y is a factor of x. Hence 15 is a
multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a
number x is always the multiple of itself and the number 1.
A prime number is a natural number that can only be divided
by 1 and itself. Its factors consist of 1 and itself only.
Prime Factorization

10. Whole numbers 1, 2, 3, 4,.. are called natural numbers and
they are used to track whole items.
If x and y are two natural numbers and x can be divided by y,
we say x is a multiple of y or y is a factor of x. Hence 15 is a
multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a
number x is always the multiple of itself and the number 1.
A prime number is a natural number that can only be divided
by 1 and itself. Its factors consist of 1 and itself only.
Hence 2, 3, 5, 7, 11, 13,… are prime numbers.
Prime Factorization

11. Whole numbers 1, 2, 3, 4,.. are called natural numbers and
they are used to track whole items.
If x and y are two natural numbers and x can be divided by y,
we say x is a multiple of y or y is a factor of x. Hence 15 is a
multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a
number x is always the multiple of itself and the number 1.
A prime number is a natural number that can only be divided
by 1 and itself. Its factors consist of 1 and itself only.
Hence 2, 3, 5, 7, 11, 13,… are prime numbers.
15 is not a prime number because 15 can be divided by 3 or 5.
Prime Factorization

12. Whole numbers 1, 2, 3, 4,.. are called natural numbers and
they are used to track whole items.
If x and y are two natural numbers and x can be divided by y,
we say x is a multiple of y or y is a factor of x. Hence 15 is a
multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a
number x is always the multiple of itself and the number 1.
A prime number is a natural number that can only be divided
by 1 and itself. Its factors consist of 1 and itself only.
Hence 2, 3, 5, 7, 11, 13,… are prime numbers.
15 is not a prime number because 15 can be divided by 3 or 5.
The number 1 is not a prime number.
Prime Factorization

13. Whole numbers 1, 2, 3, 4,.. are called natural numbers and
they are used to track whole items.
If x and y are two natural numbers and x can be divided by y,
we say x is a multiple of y or y is a factor of x. Hence 15 is a
multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a
number x is always the multiple of itself and the number 1.
A prime number is a natural number that can only be divided
by 1 and itself. Its factors consist of 1 and itself only.
Hence 2, 3, 5, 7, 11, 13,… are prime numbers.
15 is not a prime number because 15 can be divided by 3 or 5.
The number 1 is not a prime number.
Example A. List the factors and the multiples of 12.
Prime Factorization

14. Whole numbers 1, 2, 3, 4,.. are called natural numbers and
they are used to track whole items.
If x and y are two natural numbers and x can be divided by y,
we say x is a multiple of y or y is a factor of x. Hence 15 is a
multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a
number x is always the multiple of itself and the number 1.
A prime number is a natural number that can only be divided
by 1 and itself. Its factors consist of 1 and itself only.
Hence 2, 3, 5, 7, 11, 13,… are prime numbers.
15 is not a prime number because 15 can be divided by 3 or 5.
The number 1 is not a prime number.
Example A. List the factors and the multiples of 12.
The factors of 12:
The multiples of 12:
Prime Factorization

15. Whole numbers 1, 2, 3, 4,.. are called natural numbers and
they are used to track whole items.
If x and y are two natural numbers and x can be divided by y,
we say x is a multiple of y or y is a factor of x. Hence 15 is a
multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a
number x is always the multiple of itself and the number 1.
A prime number is a natural number that can only be divided
by 1 and itself. Its factors consist of 1 and itself only.
Hence 2, 3, 5, 7, 11, 13,… are prime numbers.
15 is not a prime number because 15 can be divided by 3 or 5.
The number 1 is not a prime number.
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The multiples of 12:
Prime Factorization

16. Whole numbers 1, 2, 3, 4,.. are called natural numbers and
they are used to track whole items.
If x and y are two natural numbers and x can be divided by y,
we say x is a multiple of y or y is a factor of x. Hence 15 is a
multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a
number x is always the multiple of itself and the number 1.
A prime number is a natural number that can only be divided
by 1 and itself. Its factors consist of 1 and itself only.
Hence 2, 3, 5, 7, 11, 13,… are prime numbers.
15 is not a prime number because 15 can be divided by 3 or 5.
The number 1 is not a prime number.
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The multiples of 12: 12=1x12,
Prime Factorization

17. Whole numbers 1, 2, 3, 4,.. are called natural numbers and
they are used to track whole items.
If x and y are two natural numbers and x can be divided by y,
we say x is a multiple of y or y is a factor of x. Hence 15 is a
multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a
number x is always the multiple of itself and the number 1.
A prime number is a natural number that can only be divided
by 1 and itself. Its factors consist of 1 and itself only.
Hence 2, 3, 5, 7, 11, 13,… are prime numbers.
15 is not a prime number because 15 can be divided by 3 or 5.
The number 1 is not a prime number.
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The multiples of 12: 12=1x12, 24=2x12,
Prime Factorization

18. Whole numbers 1, 2, 3, 4,.. are called natural numbers and
they are used to track whole items.
If x and y are two natural numbers and x can be divided by y,
we say x is a multiple of y or y is a factor of x. Hence 15 is a
multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a
number x is always the multiple of itself and the number 1.
A prime number is a natural number that can only be divided
by 1 and itself. Its factors consist of 1 and itself only.
Hence 2, 3, 5, 7, 11, 13,… are prime numbers.
15 is not a prime number because 15 can be divided by 3 or 5.
The number 1 is not a prime number.
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The multiples of 12: 12=1x12, 24=2x12, 36=3x12, 48, 60, etc…
Prime Factorization

19. To factor a number x means to write x as a product, i.e. write
x as a*b for some a and b.
Prime Factorization

20. To factor a number x means to write x as a product, i.e. write
x as a*b for some a and b. For example 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3.
Prime Factorization

21. To factor a number x means to write x as a product, i.e. write
x as a*b for some a and b. For example 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3.
We say the factorization is complete if all the factors are
prime numbers.
Prime Factorization

22. To factor a number x means to write x as a product, i.e. write
x as a*b for some a and b. For example 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3.
We say the factorization is complete if all the factors are
prime numbers. Hence 12=2*2*3 is factored completely,
Prime Factorization

23. To factor a number x means to write x as a product, i.e. write
x as a*b for some a and b. For example 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3.
We say the factorization is complete if all the factors are
prime numbers. Hence 12=2*2*3 is factored completely,
2*6 is not complete because 6 is not a prime number.
Prime Factorization

24. To factor a number x means to write x as a product, i.e. write
x as a*b for some a and b. For example 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3.
We say the factorization is complete if all the factors are
prime numbers. Hence 12=2*2*3 is factored completely,
2*6 is not complete because 6 is not a prime number.
Exponents
Prime Factorization

25. To factor a number x means to write x as a product, i.e. write
x as a*b for some a and b. For example 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3.
We say the factorization is complete if all the factors are
prime numbers. Hence 12=2*2*3 is factored completely,
2*6 is not complete because 6 is not a prime number.
Exponents
To simplify writing repetitive multiplication, we write 22 for 2*2,
we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.
Prime Factorization

26. To factor a number x means to write x as a product, i.e. write
x as a*b for some a and b. For example 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3.
We say the factorization is complete if all the factors are
prime numbers. Hence 12=2*2*3 is factored completely,
2*6 is not complete because 6 is not a prime number.
Exponents
To simplify writing repetitive multiplication, we write 22 for 2*2,
we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.
We write x*x*x…*x as xN where N is the number of x’s
multiplied to itself.
Prime Factorization

27. To factor a number x means to write x as a product, i.e. write
x as a*b for some a and b. For example 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3.
We say the factorization is complete if all the factors are
prime numbers. Hence 12=2*2*3 is factored completely,
2*6 is not complete because 6 is not a prime number.
Exponents
To simplify writing repetitive multiplication, we write 22 for 2*2,
we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.
We write x*x*x…*x as xN where N is the number of x’s
multiplied to itself. N is called the exponent, or the power of x.
Prime Factorization

28. To factor a number x means to write x as a product, i.e. write
x as a*b for some a and b. For example 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3.
We say the factorization is complete if all the factors are
prime numbers. Hence 12=2*2*3 is factored completely,
2*6 is not complete because 6 is not a prime number.
Exponents
To simplify writing repetitive multiplication, we write 22 for 2*2,
we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.
Example B. Calculate.
32 =
43 =
We write x*x*x…*x as xN where N is the number of x’s
multiplied to itself. N is called the exponent, or the power of x.
Prime Factorization

29. To factor a number x means to write x as a product, i.e. write
x as a*b for some a and b. For example 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3.
We say the factorization is complete if all the factors are
prime numbers. Hence 12=2*2*3 is factored completely,
2*6 is not complete because 6 is not a prime number.
Exponents
To simplify writing repetitive multiplication, we write 22 for 2*2,
we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.
Example B. Calculate.
32 = 3*3 = 9
43 =
We write x*x*x…*x as xN where N is the number of x’s
multiplied to itself. N is called the exponent, or the power of x.
Prime Factorization

30. To factor a number x means to write x as a product, i.e. write
x as a*b for some a and b. For example 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3.
We say the factorization is complete if all the factors are
prime numbers. Hence 12=2*2*3 is factored completely,
2*6 is not complete because 6 is not a prime number.
Exponents
To simplify writing repetitive multiplication, we write 22 for 2*2,
we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.
Example B. Calculate.
32 = 3*3 = 9 (note that 3*2 = 6)
43 =
We write x*x*x…*x as xN where N is the number of x’s
multiplied to itself. N is called the exponent, or the power of x.
Prime Factorization

31. To factor a number x means to write x as a product, i.e. write
x as a*b for some a and b. For example 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3.
We say the factorization is complete if all the factors are
prime numbers. Hence 12=2*2*3 is factored completely,
2*6 is not complete because 6 is not a prime number.
Exponents
To simplify writing repetitive multiplication, we write 22 for 2*2,
we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.
Example B. Calculate.
32 = 3*3 = 9 (note that 3*2 = 6)
43 = 4*4*4
We write x*x*x…*x as xN where N is the number of x’s
multiplied to itself. N is called the exponent, or the power of x.
Prime Factorization

32. To factor a number x means to write x as a product, i.e. write
x as a*b for some a and b. For example 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3.
We say the factorization is complete if all the factors are
prime numbers. Hence 12=2*2*3 is factored completely,
2*6 is not complete because 6 is not a prime number.
Exponents
To simplify writing repetitive multiplication, we write 22 for 2*2,
we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.
Example B. Calculate.
32 = 3*3 = 9 (note that 3*2 = 6)
43 = 4*4*4 = 64
We write x*x*x…*x as xN where N is the number of x’s
multiplied to itself. N is called the exponent, or the power of x.
Prime Factorization

33. To factor a number x means to write x as a product, i.e. write
x as a*b for some a and b. For example 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3.
We say the factorization is complete if all the factors are
prime numbers. Hence 12=2*2*3 is factored completely,
2*6 is not complete because 6 is not a prime number.
Exponents
To simplify writing repetitive multiplication, we write 22 for 2*2,
we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.
Example B. Calculate.
32 = 3*3 = 9 (note that 3*2 = 6)
43 = 4*4*4 = 64 (note that 4*3 = 12)
We write x*x*x…*x as xN where N is the number of x’s
multiplied to itself. N is called the exponent, or the power of x.
Prime Factorization

34. Numbers that are factored completely may be written
using the exponential notation.
Prime Factorization

35. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 =
Prime Factorization

36. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3
Prime Factorization

37. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
Prime Factorization

38. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 =
Prime Factorization

39. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25
Prime Factorization

40. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5
Prime Factorization

41. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
Prime Factorization

42. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
Each number has a unique form when its completely factored
into prime factors and arranged in order.
Prime Factorization

43. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
Each number has a unique form when its completely factored
into prime factors and arranged in order.
By “unique form” we mean
that there is only one outcome.
Prime Factorization

44. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
Each number has a unique form when its completely factored
into prime factors and arranged in order. This is analogous to
chemistry where each chemical has a unique chemical
composition of basic elements such as H20 for water.
Prime Factorization

45. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
Each number has a unique form when its completely factored
into prime factors and arranged in order. This is analogous to
chemistry where each chemical has a unique chemical
composition of basic elements such as H20 for water.
Larger numbers may be factored using a vertical format.
Prime Factorization

46. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
Each number has a unique form when its completely factored
into prime factors and arranged in order. This is analogous to
chemistry where each chemical has a unique chemical
composition of basic elements such as H20 for water.
Larger numbers may be factored using a vertical format.
Example C. Factor 144 completely. (Vertical format)
Prime Factorization

47. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
Each number has a unique form when its completely factored
into prime factors and arranged in order. This is analogous to
chemistry where each chemical has a unique chemical
composition of basic elements such as H20 for water.
Larger numbers may be factored using a vertical format.
Example C. Factor 144 completely. (Vertical format)
144
12 12
Prime Factorization

48. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
Each number has a unique form when its completely factored
into prime factors and arranged in order. This is analogous to
chemistry where each chemical has a unique chemical
composition of basic elements such as H20 for water.
Larger numbers may be factored using a vertical format.
Example C. Factor 144 completely. (Vertical format)
144
12 12
3 4 3 4
Prime Factorization

49. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
Each number has a unique form when its completely factored
into prime factors and arranged in order. This is analogous to
chemistry where each chemical has a unique chemical
composition of basic elements such as H20 for water.
Larger numbers may be factored using a vertical format.
Example C. Factor 144 completely. (Vertical format)
144
12 12
3 4 3 4
2 22 2
Prime Factorization

50. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
Each number has a unique form when its completely factored
into prime factors and arranged in order. This is analogous to
chemistry where each chemical has a unique chemical
composition of basic elements such as H20 for water.
Larger numbers may be factored using a vertical format.
Example C. Factor 144 completely. (Vertical format)
144
12 12
3 4 3 4
2 22 2
Gather all the prime numbers at
the end of the branches we have
144 = 3*3*2*2*2*2 = 3224.
Prime Factorization

51. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
Prime Factorization
Each number has a unique form when its completely factored
into prime factors and arranged in order. This is analogous to
chemistry where each chemical has a unique chemical
composition of basic elements such as H20 for water.
Larger numbers may be factored using a vertical format.
Example C. Factor 144 completely. (Vertical format)
144
12 12
3 4 3 4
2 22 2
Gather all the prime numbers at
the end of the branches we have
144 = 3*3*2*2*2*2 = 3224.
Note that we obtain the same answer
regardless how we factor at each step.

52. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
LCM

53. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example D. Find the LCM of 4 and 6.
LCM

54. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example D. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
LCM

55. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example D. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
LCM

56. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example D. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
The smallest of the common multiples is 12,
LCM

57. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example D. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
The smallest of the common multiples is 12, so LCM{4, 6 } = 12.
LCM

58. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example D. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
The smallest of the common multiples is 12, so LCM{4, 6 } = 12.
We may improve the above listing-method for finding the LCM.
LCM

59. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example D. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
The smallest of the common multiples is 12, so LCM{4, 6 } = 12.
We may improve the above listing-method for finding the LCM.
Given two or more numbers, we start with the largest number,
LCM

60. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example D. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
The smallest of the common multiples is 12, so LCM{4, 6 } = 12.
We may improve the above listing-method for finding the LCM.
Given two or more numbers, we start with the largest number,
list its multiples in order until we find the one that can divide all
the other numbers.
LCM

61. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example D. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
The smallest of the common multiples is 12, so LCM{4, 6 } = 12.
Example E. Find the LCM of 8, 9, and 12.
We may improve the above listing-method for finding the LCM.
Given two or more numbers, we start with the largest number,
list its multiples in order until we find the one that can divide all
the other numbers.
LCM

62. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example D. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
The smallest of the common multiples is 12, so LCM{4, 6 } = 12.
Example E. Find the LCM of 8, 9, and 12.
The largest number is 12 and the multiples of 12 are 12, 24,
36, 48, 60, 72, 84 …
We may improve the above listing-method for finding the LCM.
Given two or more numbers, we start with the largest number,
list its multiples in order until we find the one that can divide all
the other numbers.
LCM

63. Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
Example D. Find the LCM of 4 and 6.
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The multiples of 6 are 6, 12, 18, 24, 30,…
The smallest of the common multiples is 12, so LCM{4, 6 } = 12.
Example E. Find the LCM of 8, 9, and 12.
The largest number is 12 and the multiples of 12 are 12, 24,
36, 48, 60, 72, 84 … The first number that is also a multiple
of 8 and 9 is 72 so LCM{8, 9, 12} = 72.
We may improve the above listing-method for finding the LCM.
Given two or more numbers, we start with the largest number,
list its multiples in order until we find the one that can divide all
the other numbers.
LCM

64. But when the LCM is large, the listing method is cumbersome.
LCM

65. But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM

66. Example F. Construct the LCM of {8, 15, 18}.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM

67. To construct the LCM:
a. Factor each number completely
Example F. Construct the LCM of {8, 15, 18}.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM

68. To construct the LCM:
a. Factor each number completely
Example F. Construct the LCM of {8, 15, 18}.
Factor each number completely,
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM

69. To construct the LCM:
a. Factor each number completely
Example F. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM

70. To construct the LCM:
a. Factor each number completely
Example F. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM

71. To construct the LCM:
a. Factor each number completely
Example F. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM

72. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example F. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM

73. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
Example F. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
The prime factors appeared in the factorizations are 2, 3 and 5.
LCM

74. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
Example F. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
The prime factors appeared in the factorizations are 2, 3 and 5.
The highest degree of each prime factor are: 23,
LCM

75. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
Example F. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
The prime factors appeared in the factorizations are 2, 3 and 5.
The highest degree of each prime factor are: 23, 32
LCM

76. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
Example F. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
The prime factors appeared in the factorizations are 2, 3 and 5.
The highest degree of each prime factor are: 23, 32 and 5,
LCM

77. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
Example F. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
The prime factors appeared in the factorizations are 2, 3 and 5.
The highest degree of each prime factor are: 23, 32 and 5,
so LCM{8, 15, 18} = 23*32*5
LCM

78. To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead.
LCM
Example F. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
The prime factors appeared in the factorizations are 2, 3 and 5.
The highest degree of each prime factor are: 23, 32 and 5,
so LCM{8, 15, 18} = 23*32*5
= 8*9*5
= 360.

79. The Construction Method takes just enough of each factor
to make the LCM which "covers" all the quantities on the
list.
Methods of Finding LCM

80. The Construction Method takes just enough of each factor
to make the LCM which "covers" all the quantities on the
list. The following is an example that illustrates the
principle behind this method.
Methods of Finding LCM

81. Methods of Finding LCM
The Construction Method takes just enough of each factor
to make the LCM which "covers" all the quantities on the
list. The following is an example that illustrates the
principle behind this method.
Example C. A student wants to take enough courses so she
meets the requirement to apply to colleges A, B, and C.
The following is a table of requirements for each college,
how many years of each subject does she need?

82. Science Math. English Arts
A 2 yrs 3 yrs 3 yrs
B 3 yrs 3 yrs 2 yrs
C 2 yrs 2 yrs 4 yrs 1 yrs
Methods of Finding LCM
The Construction Method takes just enough of each factor
to make the LCM which "covers" all the quantities on the
list. The following is an example that illustrates the
principle behind this method.
Example C. A student wants to take enough courses so she
meets the requirement to apply to colleges A, B, and C.
The following is a table of requirements for each college,
how many years of each subject does she need?

83. Science Math. English Arts
A 2 yrs 3 yrs 3 yrs
B 3 yrs 3 yrs 2 yrs
C 2 yrs 2 yrs 4 yrs 1 yrs
Methods of Finding LCM
The Construction Method takes just enough of each factor
to make the LCM which "covers" all the quantities on the
list. The following is an example that illustrates the
principle behind this method.
Example C. A student wants to take enough courses so she
meets the requirement to apply to colleges A, B, and C.
The following is a table of requirements for each college,
how many years of each subject does she need?

84. Science Math. English Arts
A 2 yrs 3 yrs 3 yrs
B 3 yrs 3 yrs 2 yrs
C 2 yrs 2 yrs 4 yrs 1 yrs
So the student needs to take:
Methods of Finding LCM
The Construction Method takes just enough of each factor
to make the LCM which "covers" all the quantities on the
list. The following is an example that illustrates the
principle behind this method.
Example C. A student wants to take enough courses so she
meets the requirement to apply to colleges A, B, and C.
The following is a table of requirements for each college,
how many years of each subject does she need?
Science-3 yr Math-3yrs English-4 yrs Arts-1yr

85. Factorization
19. List the all the factors and the first 4 multiples of the
following numbers. 6, 9, 10, 15, 16, 24, 30, 36, 42, 56, 60.
Calculate.
1. 33 2. 42 3. 52 4. 53 5. 62 6. 63 7. 72
8. 82 9. 92 10. 102 11. 103 12. 104 13. 105
14. 1002 15. 1003 16. 1004 17. 112 18. 122
20. Factor completely and arrange the factors from smallest to
the largest in the exponential notation: 4, 8, 12, 16, 18, 24,
27, 32, 36, 45, 48, 56, 60, 63, 72, 75, 81, 120.

86. LCM
Find the LCM.
1. a.{6, 8} b. {6, 9} c. {3, 4}
d. {4, 10}
2. a.{5, 6, 8} b. {4, 6, 9} c. {3, 4, 5}
d. {4, 6, 10}
3. a.{6, 8, 9} b. {6, 9, 10} c. {4, 9, 10}
d. {6, 8, 10}
4. a.{4, 8, 15} b. {8, 9, 12} c. {6, 9, 15}
5. a.{6, 8, 15} b. {8, 9, 15} c. {6, 9, 16}
6. a.{8, 12, 15} b. { 9, 12, 15} c. { 9, 12, 16}
7. a.{8, 12, 18} b. {8, 12, 20} c. { 12, 15, 16}
8. a.{8, 12, 15, 18} b. {8, 12, 16, 20}
9. a.{8, 15, 18, 20} b. {9, 16, 20, 24}