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
4.2.3. 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. 228
4.2.3.2.3. A different version of stacked division – from Mrs. Bohenke’s 7 th grade math
class at Bunger Junior High in Evansdale, Iowa J
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. 230
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. 231
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. 232
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. 233
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. 234
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. 237
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
4.2.5.2. 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. 238
4.2.5.2.3. What is the largest prime number that has been found? As of printing of the
2nd book: 2 6972593 – 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. 240
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 2 p – 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
4.2.6. Problems and Exercises p. 241
4.2.6.1. Home work: 1-7, 8ab, 12-16, 18-23, 26-31