This document discusses prime factorization, greatest common factors (GCF), least common multiples (LCM), and the Euclidean algorithm. It provides examples of finding the prime factorization of numbers, using factor trees and canonical representation. It also gives steps for calculating the GCF and LCM of two or more numbers by finding their prime factorizations and examining the exponents of common factors. The Euclidean algorithm is introduced as an alternate method for finding the GCF of two numbers based on repeated division. Applications of these concepts to rational numbers and fractions are also described.

1. PRIME FACTORIZATION
GCF
LCM
RATIONAL REVISITED
EUCLIDEAN ALGORITHM
The Nature of Modern Mathematics

2. PRIME FACTORIZATION
• The question of “factoring a number” is simply one
of reversing the process of multiplication.
3 ∙ 6 = 18

3. PRIME FACTORIZATION
• It should be clear that if a number is composite, it
can be factored into two natural numbers greater
than 1. These two numbers themselves will be
prime or composite. If they are prime, then we have
a prime factorization.

4. FUNDAMENTAL THEOREM OF ARITHMETIC
• Every counting number greater than 1 is either a
prime or a product of primes, and the factorization
is unique.
• What are the possible factorizations of 18?

5. ILLUSTRATIVE EXAMPLE:
• Find the prime factorization of 1001.
By inspection
By factor tree

6. ILLUSTRATIVE EXAMPLE:
• Find the prime factorization of 1400.
By factor tree
By canonical representation

7. ILLUSTRATIVE EXAMPLE:
• Find the prime factorization of 3465.
By factor tree
By canonical representation

8. APPLICATIONS OF PRIME FACTORIZATION
• Recall that if m = 𝑑 ∙ 𝑘, then m is called a multiple
of d and k, and k are called factors of m.

9. GREATEST COMMON FACTOR (GCF)
• Find the set of common factors of 24 and 30,
Factors of 24 = { ______ }
Factors of 30 = { ______ }

10. GREATEST COMMON FACTOR (GCF)
• Find the gcf of 300, 144, and 108.
• Find the gcf of 15 and 28.
If the gcf of two numbers is 1, we say that numbers are
relatively prime.

11. GREATEST COMMON FACTOR (GCF)
• Examples: Relatively primes
15 and 33
15 and 28

12. LEAST COMMON MULTIPLE (LCM)
• Consider the set of multiples of 24 and 30:
24: { ________________}
30: { ________________}
The set of common multiples of 24 and 30 is infinite:

13. SUMMARY
• To Find the Greatest Common
Factor
1. Find the prime factorization.
2. Write in canonical form.
3. Choose the representative of
each factor with the smallest
exponent.
4. Take the product of the
representatives.
• To Find the Least Common Factor
1. Find the prime factorization.
2. Write in canonical form.
3. Choose the representative of
each factor with the largest
exponent.
4. Take the product of the
representatives.

14. GETTING GCF / LCM
• 1. What is the GCF and LCM of 10 and 12?
• 2. What is the GCF and LCM of 504 and 540?
• 3. Give the gcf and lcm of 18, 28, and 120.

15. RATIONAL NUMBERS REVISITED
• We may use the ideas of gcf and lcm when working with
rational numbers. The idea of greatest common factor, for
example, is useful when reducing or simplifying rational
expression.
• EXAMPLE: Reduce 24/30

16. RATIONAL NUMBERS REVISITED
• EXAMPLE: 1. Reduce 24/30 2. Reduce 300/144
• Solution: Note: 24 = 23
∙ 3
30 = 2 ∙ 3 ∙5
thus,
24
30
=
2∙3 ∙22
(2∙3)∙5
=
4
5

17. FUNDAMENTAL PROPERTY OF FRACTIONS
• Fundamental Property of Fractions: If a/b is any
rational number and x is any nonzero integer, then
𝒂 ∙ 𝒙
𝒃 ∙ 𝒙
=
𝒙 ∙ 𝒂
𝒙 ∙ 𝒃
=
𝒂
𝒃

18. EUCLIDEAN ALGORITHM
The gcf of two numbers can be found using another
method, which is attributed to Euclid. It is called the
Euclidean Algorithm and is based on repeated
division.
For example, find the gcf of 108 and 300.

19. EUCLIDEAN ALGORITHM
gcf = ( 300, 108 )
300 = 108 (2) + 84
108 = 84 ( 1 ) + 24
84 = 24 ( 3 ) + 12
24 = 12 ( 2 ) + 0  the last divisor is 12 and also the gcf
Therefore ( 300, 108 ) = gcf = 12.

20. OTHER WAY OF FINDING LCM
• 𝒍𝒄𝒎 ∙ 𝒈𝒄𝒇 = 𝒂 ∙ 𝒃 𝐨𝐫 𝒍𝒄𝒎 =
𝒂∙𝒃
𝒈𝒄𝒇
• For example, find the lcm of 108 and 300.
Solution:
Since (108,300) = 12
• 𝒍𝒄𝒎 =
(𝟏𝟎𝟖∙𝟑𝟎𝟎)
𝟏𝟐
=
𝟑𝟐,𝟒𝟎𝟎
𝟏𝟐
= 𝟐, 𝟕𝟎𝟎
Therefore the lcm of ( 300, 108 ) is 2,700

21. PROBLEM SET
1. What do we mean when we say the numbers are relatively?
2. Find the prime factorization of each of the following:
a. 60 b. 72 c. 95 d. 1425
3. Find the gcf and lcm of the following:
a. { 60, 72} b. { 95, 1425 } c. { 12, 52, 171 }
4. Find the gcf and lcm using Euclidean Algorithm:
a. { 357, 629} b. { 7,957, 11,023}

22. REFERENCE:
• Smith Karl J., The Nature of Modern Mathematics, Brooks/Cole Publishing
Co. California, 1973.

23. THANK YOU
www.slideshare.net/reycastro1
@reylkastro2
reylkastro