Transcript Thinking Mathematically

Thinking
Mathematically
Number Theory and the Real Number
System
5.1 Prime and Composite Numbers
The Set of Natural Numbers
N = {1,2,3,4,5,6,7, 8, 9, 10, 11, ... }.
Divisibility
If a and b are natural numbers, a is divisible
by b if the operation of dividing a by b
leaves a remainder of 0. This is the same as
saying that b is a divisor of a, or b divides
a. All three statements are symbolized by
writing b|a.
If b|a, then b is a factor of a
Rules of Divisibility
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Even numbers (last digit is even) are divisible by 2
Numbers ending in 0, 5 are divisible by 5
Numbers ending in 0 are divisible by 10
To be divisible by a composite number, must be
divisible by factor of the composite number.
• Table 5.1
Example Divisibility
Exercise Set 5.1 #5
Determine if 26,428 is divisible by each of the
following numbers:
2
3
4
5
6
8
9
10
12
Prime Numbers
A prime number is a natural number greater
than 1 that has only itself and 1 as factors.
Composite Numbers
A composite number is a natural number
greater than 1 that is divisible by a number
other than itself and 1.
The Fundamental Theorem of
Arithmetic
Every composite number can be expressed as a
product of prime numbers in one and only one
way (if the order of the factors is disregarded).
Prime factorization
The prime factors of a natural number can be found by
constructing a “factor tree.” Write the given number as a
product and continue to factor each composite number
until only prime numbers remain.
Example: Prime Factorization
Exercise Set 5.1 #33
Find the prime factorization of 663
Finding the Greatest Common
Divisor of Two or More Numbers
Using Prime Factorization
To find the greatest common divisor of two or more
numbers:
1. Write the prime factorization of each number.
2. Select each prime factor with the smallest
exponent that is common to each of the prime
factorizations.
3. Form the product of the numbers from step 2.
The greatest common divisor is the product of
these factors.
[The GCD is the intersection of the two sets of
factors]
Example: GCD
Exercise Set 5.1 #49
Find the Greatest Common Divisor of 60 and 108
Finding the Least Common Multiple
Using Prime Factorization
To find the least common multiple of two or more
numbers:
1. Write the prime factorization of each number.
2. Select every prime factor that occurs, raised to
the greatest power to which it occurs, in these
factorizations.
3. Form the product of the numbers from step 2.
The least common multiple is the product of
these factors.
[The LCM is the union of the two sets of factors]
Example: LCM
Exercise Set 5.1 #63
Find the Least Common Multiple of 72 and 120
Thinking
Mathematically
Number Theory and the Real Number
System
5.1 Prime and Composite Numbers