Prime and Composite Numbers
! A natural number that has exactly two distinct factors is called a
Section 4.2
prime number.
!
Prime and Composite Numbers
Examples: 19, 37, 53
! A natural number that has more than two distinct factors is called
a composite number.
!
Examples: 16, 75, 96
! The number 1 is neither prime nor composite.
Mathematics for Elementary School Teachers - 4th Edition
O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK
Linda Roper
1
Finding the Prime Factorization of a Number
2
Factor Trees
! When a number is expressed as a product of primes, the
expression is called the prime factorization of the number.
! Break the starting number into two factors. Then break each of
those factors into two factors, and so on, until the numbers in
the bottom row of the factor tree are prime.
! Example: 42 = 2 ! 3 ! 7; 2 ! 3 ! 7 is the prime factorization of 42
! Use exponents to represent factors that repeat.
!
Three possible factor trees for the number 30
! Each composite number can be expressed as the product of
prime numbers in exactly one way, disregarding the order of the
factors.
3
4
Stacked Division
Greatest Common Factor
! Choose the smallest prime that will divide the number. Divide by the prime
! The greatest common factor (GCF) of two natural numbers is the
and determine whether this prime will also divide the quotient. Repeat with
successive quotients as long as possible, until this prime will no longer divide
the quotient.
! Use the next smallest prime and continue the process.
greatest natural number that is a factor of both numbers.
! The greatest common factor is sometimes called the greatest
common divisor, and is represented by GCD.
! 60 = 22 ! 3 ! 5
5
Finding the GCF: Intersection of Sets
6
Finding the GCF: Prime Factorization
! List the factors of each number.
! Identify the common factors.
! Write the prime factorization of each number.
! The GCF is the product of the common prime factors.
! Select the greatest of these common factors.
!
Example: Find the GCF of 120 and 252.
120 = 2 ! 2 ! 2 ! 3 ! 5
252 = 2 ! 2 ! 3 ! 3 ! 7
The GCF is 2 ! 2 ! 3 = 12.
! Intersection of the sets.
7
8
Least Common Multiple
Finding the LCM: Intersection of Sets
! List some of the multiples of each number.
! Identify the common multiples.
! The least common multiple (LCM) of two natural
numbers is the smallest natural number that is a
multiple of both the natural numbers.
!
Intersection of Sets
!
Prime Factorization
! Select the smallest common multiple.
9
Finding the LCM: Prime Factorization
Relationships Involving GCF and LCM
! Write the prime factorization of each number.
! The LCM is the product of the common prime factors counted the
! The product of the GCF and the LCM of two numbers is the
product of the two numbers.
maximum number of times they occur in any of the numbers.
!
10
!
Example: Find the LCM of 28 and 30.
28 = 2 ! 2 ! 7
30 = 2 ! 3 ! 5
The LCM is 2 ! 2 ! 3 ! 5 ! 7 = 420.
!
Find the GCF and the LCM of 6 and 8.
Solution: The GCF is 2. The LCM is 24. The product, 2 ! 24 = 48, is
the same as the product of the two given numbers, 6 ! 8 = 48.
! Union of the sets.
11
12
Prime Numbers
! Twin primes are consecutive prime numbers with a difference
of 2.
!
Example: 3 and 5 are twin primes.
! There is no largest prime number.
The End
Section 4.2
Linda Roper
13
14