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Slide 5 - 1Copyright © 2009 Pearson Education, Inc.
5.1
Number Theory
Slide 5 - 2Copyright © 2009 Pearson Education, Inc.
Number Theory
The study of numbers and their properties. The numbers we use to count are called natural
numbers, N, or counting numbers.
{1,2,3,4,5,...}N
Slide 5 - 3Copyright © 2009 Pearson Education, Inc.
Factors
The natural numbers that are multiplied together to equal another natural number are called factors of the product.
Example: The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
Slide 5 - 4Copyright © 2009 Pearson Education, Inc.
Divisors
If a and b are natural numbers and the quotient of b divided by a has a remainder of 0, then we say that a is a divisor of b or a divides b.
Slide 5 - 5Copyright © 2009 Pearson Education, Inc.
Prime and Composite Numbers
A prime number is a natural number greater than 1 that has exactly two factors (or divisors), itself and 1.
A composite number is a natural number that is divisible by a number other than itself and 1.
The number 1 is neither prime nor composite, it is called a unit.
Slide 5 - 6Copyright © 2009 Pearson Education, Inc.
Rules of Divisibility
285The number ends in 0 or 5.
5
844 since 44 4
The number formed by the last two digits of the number is divisible by 4.
4
846 since 8 + 4 + 6 = 18
The sum of the digits of the number is divisible by 3.
3846The number is even.2
ExampleTestDivisible by
Slide 5 - 7Copyright © 2009 Pearson Education, Inc.
Create a list from 1 – 100
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100
Slide 5 - 8Copyright © 2009 Pearson Education, Inc.
The Fundamental Theorem of Arithmetic
Every composite number can be expressed as a unique product of prime numbers.
This unique product is referred to as the prime factorization of the number.
Slide 5 - 9Copyright © 2009 Pearson Education, Inc.
Finding Prime Factorizations
Branching Method: Select any two numbers whose product is
the number to be factored. If the factors are not prime numbers,
continue factoring each number until all numbers are prime.
Slide 5 - 10Copyright © 2009 Pearson Education, Inc.
Example of branching method
Therefore, the prime factorization of
3190 = 2 • 5 • 11 • 29.
3190
319 10
11 29 2 5
Slide 5 - 11Copyright © 2009 Pearson Education, Inc.
1. Divide the given number by the smallest prime number by which it is divisible.
2. Place the quotient under the given number.3. Divide the quotient by the smallest prime
number by which it is divisible and again record the quotient.
4. Repeat this process until the quotient is a prime number.
Division Method
Slide 5 - 12Copyright © 2009 Pearson Education, Inc.
Write the prime factorization of 663.
The final quotient 17, is a prime number, so we stop. The prime factorization of 663 is 3 •13 •17
Example of division method
13
3
17
221
663
Slide 5 - 13Copyright © 2009 Pearson Education, Inc.
Example 1: p. 218# 37
Slide 5 - 14Copyright © 2009 Pearson Education, Inc.
Greatest Common Factor
The greatest common factor (GCF) of a set of natural numbers is the largest natural number that divides (without remainder) every number in that set.
Slide 5 - 15Copyright © 2009 Pearson Education, Inc.
Finding the GCF of Two or More Numbers Determine the prime factorization of each
number. List each prime factor with smallest
exponent that appears in each of the prime factorizations.
Determine the product of the factors found in step 2.
Slide 5 - 16Copyright © 2009 Pearson Education, Inc.
Example (GCF)
Find the GCF of 63 and 105. 63 = 32 • 7 105 = 3 • 5 • 7
Smallest exponent of each factor:3 and 7
So, the GCF is 3 • 7 = 21.
Slide 5 - 17Copyright © 2009 Pearson Education, Inc.
Find the GCF between 36 and 54
Slide 5 - 18Copyright © 2009 Pearson Education, Inc.
Least Common Multiple
The least common multiple (LCM) of a set of natural numbers is the smallest natural number that is divisible (without remainder) by each element of the set.
Slide 5 - 19Copyright © 2009 Pearson Education, Inc.
Finding the LCM of Two or More Numbers Determine the prime factorization of each
number. List each prime factor with the greatest
exponent that appears in any of the prime factorizations.
Determine the product of the factors found in step 2.
Slide 5 - 20Copyright © 2009 Pearson Education, Inc.
Example (LCM)
Find the LCM of 63 and 105. 63 = 32 • 7105 = 3 • 5 • 7
Greatest exponent of each factor:32, 5 and 7
So, the LCM is 32 • 5 • 7 = 315.
Slide 5 - 21Copyright © 2009 Pearson Education, Inc.
Find the LCM between 36 and 54
Slide 5 - 22Copyright © 2009 Pearson Education, Inc.
Example of GCF and LCM
Find the GCF and LCM of 48 and 54. Prime factorizations of each:
48 = 2 • 2 • 2 • 2 • 3 = 24 • 354 = 2 • 3 • 3 • 3 = 2 • 33
GCF = 2 • 3 = 6
LCM = 24 • 33 = 432
Slide 5 - 23Copyright © 2009 Pearson Education, Inc.
Homework
P. 218# 15 – 54 (x3)