Sifting for Prime
Numbers
Investigating Prime and
Composite Numbers
Learning Goals
Key Terms
In this lesson, you will:
 prime numbers
 Distinguish between prime and composite  composite numbers
 multiplicative identity
numbers.
 Identify and use the multiplicative identity.
Y
ou may have heard that during the time of Columbus, many people believed that the Earth was flat. However, some historians think that this is a misconception. These historians do make a good point. Greek mathematician, geographer, and astronomer Eratosthenes (pronounced Er-uh-TOSS-thuh-neez) has been credited as the first person to calculate the circumference of the Earth. He made this calculation roughly around 240 BCE — thousands of years before Columbus’ time! If you remember, circumference is the distance around a circle. Therefore, even back then, people assumed that the Earth was round. So, where do you think this misconception that many people in the 1400s believed © 2011 Carnegie Learning
© 2011 Carnegie Learning
in a flat Earth came from? 1.3 Investigating Prime and Composite Numbers • 31
1.3 Investigating Prime and Composite Numbers • 31
Problem 1
Students explore the number 1.
The number 1 is not a prime
number nor is it a composite
number. It is a factor of every
number, and the multiplicative
identity.
Problem 1
The Game
A sieve is an old tool that is used to separate small particles from larger particles and is
usually a box with a screen for a bottom that allows the smaller pieces to fall through.
The Sieve of Eratosthenes screens out all of the composite numbers and leaves only the
prime numbers. Primenumbers are numbers greater than 1 with exactly two distinct
factors, 1 and the number itself. Compositenumbers are numbers that have more than
Grouping
Have students complete
Questions 1 through 5 with
a partner. Then share the
responses as a class.
two distinct factors. You and your partner will use the Sieve of Eratosthenes to determine
all of the prime numbers up to 100.
The figure shows the first 100 numbers written in numerical order in an array.
12345678910
11121314151617181920
21222324252627282930
Share Phase,
Questions 1 through 5
• Why does the sieve work?
• Why do you think
31323334353637383940
41424344454647484950
51525354555657585960
61626364656667686970
Eratosthenes called his tool
a sieve?
71727374757677787980
81828384858687888990
• Why can you eliminate all
even numbers except 2
as primes?
• How could you check to see
that the sieve found all the
prime numbers under 100?
Which operations would
you use?
1. Start by putting a square around the number 1 because it is not a prime or composite
number.
2. Circle the number 2 and cross out all of the multiples of 2.
3. Circle the next number after 2 that is not crossed out. Then cross out all the multiples
of that number that are not already crossed out.
4. Continue in this fashion until you come to the first number greater than 10 that is not
crossed out. All of the remaining numbers have “fallen through the sieve” and are
prime numbers.
5. List all of the prime numbers up to 100.
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,
89,97
32 • Chapter 1 Factors, Multiples, Primes, and Composites
32 • Chapter 1 Factors, Multiples, Primes, and Composites
© 2011 Carnegie Learning
in making the sieve work?
919293949596979899100
© 2011 Carnegie Learning
• What is the role of multiples
Grouping
Have students complete
Questions 6 through 10 with
a partner. Then share the
responses as a class.
6. How many of the prime numbers are even?
Onlyoneevennumberisprime.Twoistheonlyevenprimenumber.
Share Phase,
Questions 6 through 10
7. Is it possible that there are even prime numbers greater than 100? Explain your
reasoning.
No.Itisnotpossiblebecauseeveryevennumbergreaterthan2isdivisibleby2,or
Why are some odd numbers
primes, while others are not?
has2asafactor,makingitacompositenumber.
8. Why did you stop at 10? How do you know that any remaining number less than 100
must be a prime number?
Idon’tneedtogohigherthan10becauseanynumberthatisdivisiblebyanother
numberwillbedivisiblebyanumberlessthan10.Numbersgreaterthan10thatdo
nothavefactorslessthan10willhavefactorsof1andthenumberitself.That
meansthatthesenumbersareprimenumbers.
9. Are all odd numbers prime? Explain your reasoning.
No.Someoddnumbershavefactorsotherthan1andthemselves.Forexample,
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9hasthefactorsof1,3,and9;therefore,itisnotaprimenumber.
10.
Recall the distinct area models you created for Speedy Builders in the previous lesson.
Identify the prime numbers. How are all the area models of prime numbers similar?
TheprimenumbersfromtheareamodelsIcreatedare2,3,5,7,11,13,17,19,23,
and29.Thesenumbersalwayshaveonlyoneareamodelbecausetheonlyfactors
ofaprimenumberare1andthenumberitself.Therearenootherwaystomake
© 2011 Carnegie Learning
otherareamodels.
1.3 Investigating Prime and Composite Numbers • 33
1.3 Investigating Prime and Composite Numbers • 33
Talk the Talk
The multiplicative identify is
defined as the number 1.
Students state the
characteristics of prime and
composite numbers.
Talk the Talk
●
The number 1 is neither prime nor composite.
●
The number 1 is a factor of every number.
The number 1 is called the multiplicative identity. Themultiplicativeidentity is the
Grouping
Ask a student to read the
information aloud. Discuss
the definition and complete
Questions 1 through 3 as
a class.
Discuss Phase,
Talk the Talk
• Besides the number 1,
number 1. When it is multiplied by a second number, the product is the second number.
An example is 1  5 5 5.
1. Explain why the number 1 is neither prime nor composite.
Thenumber1isnotprimebecauseitdoesnothavetwodistinctfactors:1and
itself.Itisnotcompositebecauseitdoesnothavemorethantwofactors.Onehas
onlyitselfasafactor,soitisneitherprimenorcomposite.
2. State the characteristics prime numbers share.
Allprimenumbershaveexactlytwodistinctfactors:1andthenumberitself.
are any other numbers
considered neither prime
nor composite?
• Can any shortcuts be used
to determine if a number is a
prime number?
3. State the characteristics composite numbers share.
Allcompositenumbershavemorethantwodistinctfactors.
Be prepared to share your solutions and methods.
34 • Chapter 1 Factors, Multiples, Primes, and Composites
34 • Chapter 1 Factors, Multiples, Primes, and Composites
© 2011 Carnegie Learning
automatically be eliminated,
without any factoring,
when searching for
prime numbers?
© 2011 Carnegie Learning
• What numbers can