Chapter 4
Number Theory
Activity: Rectangle Dimensions
y
Section 4.2, page 221
Prime and Composite Numbers
NUMBER
DIMENSIONS OF
RECTANGLES
FACTORS
PRIME or
COMPOSITE?
Using the Dimensions of Rectangles
Chart, use Snap-Cubes to create as
many rectangles as possible using
the number of cubes listed in the
left-hand column.
Comparing
Prime and Composite Numbers
12
A natural number that has exactly two
distinct factors is called a _________
number.
y A natural number that has more than
two distinct factors is called a _________
number.
y Note: The number 1 has only one
distinct factor, so it is _________ prime
nor composite.
9
y
10
2
5
7
Activity:
Identifying Prime Numbers
Sieve of Erastosthenes: used to
identify prime numbers
y Using the list of numbers from 1100,
00, cross
c oss out all
a multiples
u t p es of
o 2
(except 2 itself).
y Do the same for multiples of 3, 5,
and 7.
y What do you discover about the
numbers that remain?
y
SIEVE
of
E
R
A
S
T
O
S
T
H
E
N
E
S
1
Using the Calculator to
Determine if Numbers are Prime
y
To determine if larger numbers are
prime, take the ________ _______ of the
number, then check to see if those
_________ numbers less than or equal
to the square root are factors. (It’s
only necessary to check the prime
numbers as factors because all other
whole numbers are multiples of
primes.)
Fundamental Theorem of
Arithmetic
Sometimes referred to as the
Unique Factorization Theorem.
y The Fundamental Theorem of
Arithmetic
t
et c states tthat
at each
eac
composite number can be expressed
as the _________ of prime numbers
in exactly one way, disregarding the
order of the factors.
y
Examples
1)
Find the prime factorization of 84 using:
b) stacked division
Using the Calculator to
Determine if Numbers are Prime
y
Example: Determine if 367 is prime or
composite.
Examples
1) Find the prime factorization of 84 using:
a) factor tree
Examples
2) Find the prime factorization of 150.
2
Greatest Common Factor,
page 229
Example of Euclidean Algorithm
Find the GCF of 253 & 322.
The greatest common factor (GCF)
of two natural numbers is the
greatest natural number that is a
factor of both numbers.
numbers
y Three Methods for Finding GCF
1. Prime Factorization
2. _________ Diagram
3. Euclidean _________ (pg. 232)
y
Examples
Examples
1) Determine the GCF of 84 and 150
by using the prime factorization of
each.
2) Determine the GCF of 24 and 32 by
listing the factors and using set
notation in conjunction with a Venn
diagram.
g
Examples
Least Common Multiple
3) Determine the GCF of 60 and 140
using the Euclidean algorithm.
y
y
The least common multiple (LCM)
of two natural numbers is the
_________ natural number that is a
multiple
p of both the natural
numbers.
3
Examples
1)
y
Examples
Use the intersection of sets to find
the LCM of 3 and 8.
2) Use prime factorization to find the
LCM of 9 and 12.
Important Reminders
Twin Primes and Relative Primes
When using prime factorization to
find the GCF and LCM:
◦ The GCF is found by using the
_________ exponent for each shared
prime power.
◦ The LCM is found by using the
_________ exponent for each prime
power.
y
Twin primes: any two _________
primes that differ by 2
Examples: 3, 5 5, 7 11, 13
y
Two numbers a and b are _________
prime iff GCF (a,b) = 1.
The GCF
GCF--LCM Product Theorem
The product of the GCF and the
LCM of two numbers is the product
of the two numbers.
y Example:
a p e: Find
d the
t e GCF
GC and
a d LCM
C of
o
12 and 16. Compare the product of
the GCF and LCM to the product of
the numbers.
Common Multiples and
Common Factors
y
y
y
Suppose S = NA where N is a whole number.
We say that S is a _________ of A (and a
multiple of N) and that N and A are
_________ of S.
How can we recognize that we are
concerned with multiples?
In terms of the meaning
g of
multiplication/division, S is the total of N
groups of size A. So if we want to find _______
of a number of groups of a certain size
(repeating groups of the same size), we are
looking for a multiple of that size. Also, you
can recognize finding multiples if you are
asked to find a ______ that can be formed
into a certain number of groups.
Developing Conceptual Understanding for Teaching Elementary Mathematics, University of Alabama
4
Common Multiples and
Common Factors
y
How can we recognize that we are
concerned with factors?
In terms of the meaning of
multiplication/division, if we are
starting
sta
t g with
w t a total
tota (S) and
a d looking
oo g
for the size of a _________ (A) or the
number of _________(N), we are
looking for factors.
Developing Conceptual Understanding for Teaching Elementary Mathematics, University of Alabama
Activity:
Common Multiples and
Common Factors
For each of the following three
problems, read carefully and first
explain if you are looking for multiples
or factors and how you know.
y Then solve the problem, describing
your reasoning.
y
Developing Conceptual Understanding for Teaching Elementary Mathematics, University of Alabama
Activity:
Common Multiples and Common Factors
Activity:
Common Multiples and Common Factors
1) Pencils are packaged 12 pencils to a
box. Erasers are packaged 18 to a
box.You wish to buy just enough
packages of each so that you can
form complete sets of one pencil and
one eraser (with nothing left over).
How many packages of each will you
buy? How many complete sets of
erasers and pencils will you have?
2) Patty is making up candy packets. She has a
box containing 24 gum drops and a bag
containing 16 pieces of bubble gum. She want
to divide up the candy into packets so that
there is only one kind of candy in a packet,
packet
each packet contains the same number of
pieces, and all the candy is used up. Suppose
further that she wants as many pieces in each
packet as possible. How many pieces of candy
are in each packet? How many packets can she
make up?
Developing Conceptual Understanding for Teaching Elementary Mathematics, University of Alabama
Activity:
Common Multiples and Common Factors
3) You have a large candy bar. You want to
divide it into equal size pieces so that if 8
children wanted to share the candy bar
equally, each child would get a whole
number of pieces and if 6 children wanted
to share the candy bar equally, each child
would get a whole number of pieces. How
many pieces would you need to divide the
candy bar into, assuming you wanted the
pieces to be as large as possible?
Developing Conceptual Understanding for Teaching Elementary Mathematics, University of Alabama
Recap
y
LEAST COMMON MULTIPLE of
numbers: The smallest number that is a
multiple of each of the given numbers.
y
GREATEST COMMON FACTOR of
numbers: The largest number that is a
factor of each of the given numbers.
Developing Conceptual Understanding for Teaching Elementary Mathematics, University of Alabama
5