Chapter 4: Number Theory
4.2
Prime and
Composite Numbers
4.2.1. Defining
Prime and Composite Numbers
4.2.1.1.
Definition of Prime and Composite
Numbers: A natural number that has
exactly two distinct factors is called a prime number. A natural number that has more than two distinct
factors is called a composite number
4.2.1.1.1. The number ONE is NOT prime or composite because it has only one
distinct factor
4.2.2. Techniques
for finding prime numbers
4.2.2.1.
Sieve of Erastosthenes – Greek mathematician (200 B.C.)
4.2.2.2.
Worksheet in
folder – let’s use the sieve to find all the primes between 1 and 100
4.2.2.3.
Your turn p. 209: Do the practice and
the reflect
4.2.3. The
role of prime numbers in mathematics
4.2.3.1.
Prime numbers as building blocks
4.2.3.1.1. Prime numbers are building blocks for composite
numbers
4.2.3.1.2. Fundamental Theorem of
Arithmetic (Unique
Factorization Theorem): Each composite number can be expressed as the product
of prime numbers in exactly one way, disregarding the order of the factors
4.2.3.2.
Finding the prime factorization of a number
4.2.3.2.1. When a number is expressed as a product of only prime
numbers it is called the prime
factorization of that number
4.2.3.2.2. Stacked division – see example p. 213
4.2.3.2.3. A different version of stacked division – from Mrs. Bohenke’s 7th grade math class at Bunger Junior
High in
4.2.3.2.4.
Your turn p. 213: Do the practice and
the reflect
4.2.4. Greatest
common factor (GCF) and least common multiple (LCM)
4.2.4.1.
Greatest common factor
4.2.4.1.1. GCF
4.2.4.1.2. Definition
of Greatest Common Factor: The
greatest common factor of two natural numbers is the greatest natural number
that is a factor of both numbers
4.2.4.1.3.
Using the intersection of sets to find the GCF
4.2.4.1.3.1.
Find all of the
factors for each of the numbers being compared
4.2.4.1.3.2.
Identify the
factors that are common to each number being compared – the intersection of the
sets of factors being compared
4.2.4.1.3.3.
The largest number
in the intersection is the GCF
4.2.4.1.3.4.
See example p.
215
4.2.4.1.4.
Using prime factorization to find the GCF
4.2.4.1.4.1.
Find the prime
factors for each of the numbers being compared
4.2.4.1.4.2.
Identify the
prime factors that are common to each number being compared
4.2.4.1.4.3.
Multiply out the
common prime factors to obtain the GCF
4.2.4.1.4.4.
See example p.
215
4.2.4.1.4.5.
Your turn p. 215: Do the practice and
the reflect
4.2.4.1.5. The Euclidean algorithm to find the GCF
4.2.4.1.5.1.
Start with two
natural numbers a and b, where a > b
4.2.4.1.5.2.
Divide a by b
4.2.4.1.5.3.
Is the remainder
zero?
4.2.4.1.5.3.1. Yes – the last divisor is the GCF
4.2.4.1.5.3.2. No – divide b by the last divisor
4.2.4.1.5.3.3. Repeat “No” until get “Yes”
4.2.4.1.5.4.
See example p.
216
4.2.4.1.5.5.
Your turn p. 216: Do the practice and the reflect
4.2.4.2.
Least Common Multiple
4.2.4.2.1. LCM
4.2.4.2.2. Definition
of the Least Common Multiple: The
least common multiple of two natural numbers is the smallest natural number
that is a multiple of both the natural numbers
4.2.4.2.3.
Using the intersection of sets to find the LCM
4.2.4.2.3.1.
Find several of
the multiples for each of the numbers being compared
4.2.4.2.3.2.
Identify the
multiples that are common to each number being compared – the intersection of
the sets of multiples being compared
4.2.4.2.3.3.
The smallest
number in the intersection is the LCM
4.2.4.2.3.4.
See example p.
218
4.2.4.2.4.
Using prime factorization to find the LCM
4.2.4.2.4.1.
Find the prime
factors for each of the numbers being compared
4.2.4.2.4.2.
Identify the highest
exponential form of all prime factors in all the numbers being compared
4.2.4.2.4.3.
Multiply out these
prime factors to obtain the LCM
4.2.4.2.4.4.
See example p.
218
4.2.4.2.4.5.
Your turn p. 218: Do the practice and
the reflect
4.2.4.3.
Dr. Moch’s trick
for finding GCF and LCM J
4.2.5. Prime
and composite numbers and relationships
4.2.5.1.
Relationships involving GCF and LCM
4.2.5.1.1. Complete the table on p. 220
4.2.5.1.2. Theorem:
The GCF-LCM Product – The product of
the GCF and the LCM of two numbers is the product of the two numbers
4.2.5.1.2.1.
Your turn p. 220: Do the practice and the reflect
4.2.5.2.
Relationships and patterns involving prime numbers
4.2.5.2.1. twin prime numbers – consecutive prime numbers with a
difference of 2
4.2.5.2.2. Is there a largest prime number? No – see p. 222
4.2.5.2.3. What is the largest prime number that has been found?
As of printing of the 2nd book: 26972593 – 1. A larger one has since been found: 220996011
– 1.
4.2.5.2.4. Can two primes have an unlimited number of composite
numbers between them? Yes – see p. 222
4.2.5.2.5. Are there still some unsolved problems in number
theory? Absolutely!
4.2.5.2.5.1.
Conjectures
still out there
4.2.5.2.5.1.1. Every even number greater than 2 is the sum of two
primes
4.2.5.2.5.1.2. There are an infinite number of pairs of primes whose
difference is 2
4.2.5.2.5.1.3. There is no odd perfect number
4.2.5.2.5.1.4. There are infinitely many primes in the form 2p
– 1
4.2.5.2.5.1.5. There are infinitely many perfect numbers
4.2.5.2.5.2.
Fermat’s last
theorem recently proved after 350 years