Thinking Mathematically

Robert F. Blitzer

Chapter 5

Number Theory and the Real Number System - all with Video Answers

Educators

Section 1

Number Theory: Prime and Composite Numbers

Problem 1

Use rules of divisibility to determine whether each number given in Exercises 1-10 is divisible by
a. 2
b. 3
c. 4
d. 5
e. 6
f. 8
g. 9
h. 10
i. 12 .
6944

Reethika Digumarthy

Reethika Digumarthy

Numerade Educator

Problem 2

Use rules of divisibility to determine whether each number given in divisible by
a. 2
b. 3
c. 4
d. 5
e. 6
f. 8
g. 9
h. 10
i. 12 .
7245

Reethika Digumarthy

Reethika Digumarthy

Numerade Educator

Problem 3

Use rules of divisibility to determine whether each number given in divisible by
a. 2
b. 3
c. 4
d. 5
e. 6
f. 8
g. 9
h. 10
i. 12 .
21,408

Reethika Digumarthy

Reethika Digumarthy

Numerade Educator

Problem 4

Use rules of divisibility to determine whether each number given in divisible by
a. 2
b. 3
c. 4
d. 5
e. 6
f. 8
g. 9
h. 10
i. 12 .
25,025

Reethika Digumarthy

Reethika Digumarthy

Numerade Educator

Problem 5

Use rules of divisibility to determine whether each number given in divisible by
a. 2
b. 3
c. 4
d. 5
e. 6
f. 8
g. 9
h. 10
i. 12 .
26,428

Reethika Digumarthy

Reethika Digumarthy

Numerade Educator

Problem 6

Use rules of divisibility to determine whether each number given in divisible by
a. 2
b. 3
c. 4
d. 5
e. 6
f. 8
g. 9
h. 10
i. 12 .
89,001

Reethika Digumarthy

Reethika Digumarthy

Numerade Educator

Problem 7

Use rules of divisibility to determine whether each number given in divisible by
a. 2
b. 3
c. 4
d. 5
e. 6
f. 8
g. 9
h. 10
i. 12 .
374,832

Reethika Digumarthy

Reethika Digumarthy

Numerade Educator

Problem 8

Use rules of divisibility to determine whether each number given in divisible by
a. 2
b. 3
c. 4
d. 5
e. 6
f. 8
g. 9
h. 10
i. 12 .
347,712

Reethika Digumarthy

Reethika Digumarthy

Numerade Educator

Problem 9

Use rules of divisibility to determine whether each number given in divisible by
a. 2
b. 3
c. 4
d. 5
e. 6
f. 8
g. 9
h. 10
i. 12 .
$6,126,120$

Reethika Digumarthy

Reethika Digumarthy

Numerade Educator

Problem 10

Use rules of divisibility to determine whether each number given in divisible by
a. 2
b. 3
c. 4
d. 5
e. 6
f. 8
g. 9
h. 10
i. 12 .
$5,941,221$

Reethika Digumarthy

Reethika Digumarthy

Numerade Educator

Problem 11

In Exercises 11-24, use a calculator to determine whether each statement is true or false. If the statement is true, explain why this is so using one of the rules of divisibility in Table 5.1 on page $253 .$
$3 \mid 5958$

Garrett Bess

Numerade Educator

Problem 12

Use a calculator to determine whether each statement is true or false. If the statement is true, explain why this is so using one of the rules of divisibility in Table 5.1 on page $253 .$
$3 \mid 8142$

Anh Nguyen

Numerade Educator

Problem 13

Use a calculator to determine whether each statement is true or false. If the statement is true, explain why this is so using one of the rules of divisibility in Table 5.1 on page $253 .$
$4 \mid 10,612$

Anh Nguyen

Numerade Educator

Problem 14

Use a calculator to determine whether each statement is true or false. If the statement is true, explain why this is so using one of the rules of divisibility in Table 5.1 on page $253 .$
$4 \mid 15,984$

Anh Nguyen

Numerade Educator

Problem 15

Use a calculator to determine whether each statement is true or false. If the statement is true, explain why this is so using one of the rules of divisibility in Table 5.1 on page $253 .$
$5 \mid 38,814$

Anh Nguyen

Numerade Educator

Problem 16

Use a calculator to determine whether each statement is true or false. If the statement is true, explain why this is so using one of the rules of divisibility in Table 5.1 on page $253 .$
$5 \mid 48,659$

Anh Nguyen

Numerade Educator

Problem 17

Use a calculator to determine whether each statement is true or false. If the statement is true, explain why this is so using one of the rules of divisibility in Table 5.1 on page $253 .$
$6 \mid 104,538$

Anh Nguyen

Numerade Educator

Problem 18

Use a calculator to determine whether each statement is true or false. If the statement is true, explain why this is so using one of the rules of divisibility in Table 5.1 on page $253 .$
$6 \mid 163,944$

Anh Nguyen

Numerade Educator

Problem 19

Use a calculator to determine whether each statement is true or false. If the statement is true, explain why this is so using one of the rules of divisibility in Table 5.1 on page $253 .$
$8 \mid 20,104$

Anh Nguyen

Numerade Educator

Problem 20

Use a calculator to determine whether each statement is true or false. If the statement is true, explain why this is so using one of the rules of divisibility in Table 5.1 on page $253 .$
$8 \mid 28,096$

Anh Nguyen

Numerade Educator

Problem 21

Use a calculator to determine whether each statement is true or false. If the statement is true, explain why this is so using one of the rules of divisibility in Table 5.1 on page $253 .$
$9 \mid 11,378$

Anh Nguyen

Numerade Educator

Problem 22

Use a calculator to determine whether each statement is true or false. If the statement is true, explain why this is so using one of the rules of divisibility in Table 5.1 on page $253 .$
$9 \mid 23,772$

Anh Nguyen

Numerade Educator

Problem 23

Use a calculator to determine whether each statement is true or false. If the statement is true, explain why this is so using one of the rules of divisibility in Table 5.1 on page $253 .$
$12 \mid 517,872$

Anh Nguyen

Numerade Educator

Problem 24

Use a calculator to determine whether each statement is true or false. If the statement is true, explain why this is so using one of the rules of divisibility in Table 5.1 on page $253 .$
$12 \mid 785,172$

Anh Nguyen

Numerade Educator

Problem 25

In Exercises 25-44, find the prime factorization of each composite number.
75

Meredith Moody

Numerade Educator

Problem 26

Find the prime factorization of each composite number.
45

Ashley Hanson

Numerade Educator

Problem 27

Find the prime factorization of each composite number.
56

Ashley Hanson

Numerade Educator

Problem 28

Find the prime factorization of each composite number.
48

Amy Jiang

Numerade Educator

Problem 29

Find the prime factorization of each composite number.
105

James Kiss

Numerade Educator

Problem 30

Find the prime factorization of each composite number.
180

Tiffany Tran

Numerade Educator

Problem 31

Find the prime factorization of each composite number.
500

Nick Johnson

Numerade Educator

Problem 32

Find the prime factorization of each composite number.
360

Cora Crumley

Numerade Educator

Problem 33

Find the prime factorization of each composite number.
663

Amy Jiang

Numerade Educator

Problem 34

Find the prime factorization of each composite number.
510

Hannah Laper

Numerade Educator

Problem 35

Find the prime factorization of each composite number.
885

Amy Jiang

Numerade Educator

Problem 36

Find the prime factorization of each composite number.
999

Amy Jiang

Numerade Educator

Problem 37

Find the prime factorization of each composite number.
1440

Meredith Moody

Numerade Educator

Problem 38

Find the prime factorization of each composite number.
1280

Hannah Laper

Numerade Educator

Problem 39

Find the prime factorization of each composite number.
1996

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 40

Find the prime factorization of each composite number.
1575

Amy Jiang

Numerade Educator

Problem 41

Find the prime factorization of each composite number.
3675

Victor Salazar

Numerade Educator

Problem 42

Find the prime factorization of each composite number.
8316

Victor Salazar

Numerade Educator

Problem 43

Find the prime factorization of each composite number.
85,800

Victor Salazar

Numerade Educator

Problem 44

Find the prime factorization of each composite number.
30,600

Victor Salazar

Numerade Educator

Problem 45

In Exercises 45-56, find the greatest common divisor of the numbers.
42 and 56

Julie Silva

Numerade Educator

Problem 46

Find the greatest common divisor of the numbers.
25 and 70

Vysakh M

Numerade Educator

Problem 47

Find the greatest common divisor of the numbers.
16 and 42

Amy Jiang

Numerade Educator

Problem 48

Find the greatest common divisor of the numbers.
66 and 90

Vysakh M

Numerade Educator

Problem 49

Find the greatest common divisor of the numbers.
60 and 108

Amy Jiang

Numerade Educator

Problem 50

Find the greatest common divisor of the numbers.
96 and 212

Amy Jiang

Numerade Educator

Problem 51

Find the greatest common divisor of the numbers.
72 and 120

Ashley Volpe

Numerade Educator

Problem 52

Find the greatest common divisor of the numbers.
220 and 400

Vysakh M

Numerade Educator

Problem 53

Find the greatest common divisor of the numbers.
342 and 380

Amy Jiang

Numerade Educator

Problem 54

Find the greatest common divisor of the numbers.
224 and 430

Vysakh M

Numerade Educator

Problem 55

Find the greatest common divisor of the numbers.
240 and 285

Ashley Volpe

Numerade Educator

Problem 56

Find the greatest common divisor of the numbers.
150 and 480

Ashley Volpe

Numerade Educator

Problem 57

In Exercises 57-68, find the least common multiple of the numbers.
42 and 56

Victor Salazar

Numerade Educator

Problem 58

Find the least common multiple of the numbers.
25 and 70

Julie Silva

Numerade Educator

Problem 59

Find the least common multiple of the numbers.
16 and 42

Victor Salazar

Numerade Educator

Problem 60

Find the least common multiple of the numbers.
66 and 90

Victor Salazar

Numerade Educator

Problem 61

Find the least common multiple of the numbers.
60 and 108

Victor Salazar

Numerade Educator

Problem 62

Find the least common multiple of the numbers.
96 and 212

Victor Salazar

Numerade Educator

Problem 63

Find the least common multiple of the numbers.
72 and 120

Ashley Volpe

Numerade Educator

Problem 64

Find the least common multiple of the numbers.
220 and 400

Julie Silva

Numerade Educator

Problem 65

Find the least common multiple of the numbers.
342 and 380

Julie Silva

Numerade Educator

Problem 66

Find the least common multiple of the numbers.
224 and 430

Julie Silva

Numerade Educator

Problem 67

Find the least common multiple of the numbers.
240 and 285

Julie Silva

Numerade Educator

Problem 68

Find the least common multiple of the numbers.
150 and 480

Julie Silva

Numerade Educator

Problem 69

In Exercises 69-74, determine all values of $d$ that make each statement true.
$9 \mid 12,34 d$

Michael Cooper

Numerade Educator

Problem 70

Determine all values of $d$ that make each statement true.
$9 \mid 23,42 d$

Mitchell Cutler

Mitchell Cutler

Numerade Educator

Problem 71

Determine all values of $d$ that make each statement true.
$8 \mid 76,523,45 d$

Nick Johnson

Numerade Educator

Problem 72

Determine all values of $d$ that make each statement true.
$8 \mid 88,888,82 d$

Nick Johnson

Numerade Educator

Problem 73

Determine all values of $d$ that make each statement true.
$4 \mid 963,23 d$

Amy Jiang

Numerade Educator

Problem 74

Determine all values of $d$ that make each statement true.
$4 \mid 752,67 d$

Amy Jiang

Numerade Educator

Problem 75

A perfect number is a natural number that is equal to the sum of its factors, excluding the number itself. In Exercises 75-78, determine whether or not each number is perfect.
28

Amy Jiang

Numerade Educator

Problem 76

A perfect number is a natural number that is equal to the sum of its factors, excluding the number itself. Determine whether or not each number is perfect.
6

Amy Jiang

Numerade Educator

Problem 77

A perfect number is a natural number that is equal to the sum of its factors, excluding the number itself. Determine whether or not each number is perfect.
20

Amy Jiang

Numerade Educator

Problem 78

A perfect number is a natural number that is equal to the sum of its factors, excluding the number itself. Determine whether or not each number is perfect.
50

Amy Jiang

Numerade Educator

Problem 79

A prime number is an emirp ("prime" spelled backward) if it becomes a different prime number when its digits are reversed. In Exercises 79-82, determine whether or not each prime number is an emirp.
41

Jacob Shpiece

Numerade Educator

Problem 80

A prime number is an emirp ("prime" spelled backward) if it becomes a different prime number when its digits are reversed.Determine whether or not each prime number is an emirp.
43

Jacob Shpiece

Numerade Educator

Problem 81

A prime number is an emirp ("prime" spelled backward) if it becomes a different prime number when its digits are reversed.Determine whether or not each prime number is an emirp.
107

Jacob Shpiece

Numerade Educator

Problem 82

A prime number is an emirp ("prime" spelled backward) if it becomes a different prime number when its digits are reversed.Determine whether or not each prime number is an emirp.
113

Jacob Shpiece

Numerade Educator

Problem 83

A prime number $p$ such that $2 p+1$ is also a prime number is called a Germain prime, named after the German mathematician Sophie Germain (1776-1831), who made major contributions to number theory. In Exercises 83-86, determine whether or not each prime number is a Germain prime.
13

Charles Carter

Numerade Educator

Problem 84

A prime number $p$ such that $2 p+1$ is also a prime number is called a Germain prime, named after the German mathematician Sophie Germain (1776-1831), who made major contributions to number theory. Determine whether or not each prime number is a Germain prime.
11

Bobby Barnes

University of North Texas

Problem 85

A prime number $p$ such that $2 p+1$ is also a prime number is called a Germain prime, named after the German mathematician Sophie Germain (1776-1831), who made major contributions to number theory. Determine whether or not each prime number is a Germain prime.
241

Jonathan Zhang

Numerade Educator

Problem 86

A prime number $p$ such that $2 p+1$ is also a prime number is called a Germain prime, named after the German mathematician Sophie Germain (1776-1831), who made major contributions to number theory. Determine whether or not each prime number is a Germain prime.
97

Jonathan Zhang

Numerade Educator

Problem 87

Find the product of the greatest common divisor of 24 and 27 and the least common multiple of 24 and 27. Compare this result to the product of 24 and 27 . Write a conjecture based on your observation.

Ashley Hanson

Numerade Educator

Problem 88

Find the product of the greatest common divisor of 48 and 72 and the least common multiple of 48 and 72. Compare this result to the product of 48 and 72 . Write a conjecture based on your observation.

Amy Jiang

Numerade Educator

Problem 89

In Carl Sagan's novel Contact, Ellie Arroway, the book's heroine, has been working at SETI, the Search for Extraterrestrial Intelligence, listening to the crackle of the cosmos. One night, as the radio telescopes are turned toward Vega, they suddenly pick up strange pulses through the background noise. Two pulses are followed by a pause, then three pulses, five, seven,
continuing through 97 . Then it starts all over again. Ellie is convinced that only intelligent life could generate the structure in the sequence of pulses. "It's hard to imagine some radiating plasma sending out a regular set of mathematical signals like this." What is it about the structure of the pulses that the book's heroine recognizes as the sign of intelligent life? Asked in another way, what is significant about the numbers of pulses?

Courtney Burson

Courtney Burson

Numerade Educator

Problem 90

There are two species of insects, Magicicada septendecim and Magicicada tredecim, that live in the same environment. They have a life cycle of exactly 17 and 13 years, respectively. For all but their last year, they remain in the ground feeding on the sap of tree roots. Then, in their last year, they emerge en masse from the ground as fully formed cricketlike insects, taking over the forest in a single night. They chirp loudly, mate, eat, lay eggs, then die six weeks later.
a. Suppose that the two species have life cycles that are not prime, say 18 and 12 years, respectively. List the set of multiples of 18 that are less than or equal to 216 . List the set of multiples of 12 that are less than or equal to 216 . Over a 216-year period, how many times will the two species emerge in the same year and compete to share the forest?
b. Recall that both species have evolved prime-number life cycles, 17 and 13 years, respectively. Find the least common multiple of 17 and 13 . How often will the two species have to share the forest?
c. Compare your answers to parts (a) and (b). What explanation can you offer for each species having a prime number of years as the length of its life cycle?

Nick Johnson

Numerade Educator

Problem 91

A relief worker needs to divide 300 bottles of water and 144 cans of food into groups that each contain the same number of items. Also, each group must have the same type of item (bottled water or canned food). What is the largest number of relief supplies that can be put in each group?

Brandon Fox

Numerade Educator

Problem 92

A choral director needs to divide 180 men and 144 women into all-male and all-female singing groups so that each group has the same number of people. What is the largest number of people that can be placed in each singing group?

Jonathan Zhang

Numerade Educator

Problem 93

You have in front of you 310 five-dollar bills and 460 tendollar bills. Your problem: Place the five-dollar bills and the ten-dollar bills in stacks so that each stack has the same number of bills, and each stack contains only one kind of bill (five-dollar or ten-dollar). What is the largest number of bills that you can place in each stack?

Derek Follett

Numerade Educator

Problem 94

Harley collects sports cards. He has 360 football cards and 432 baseball cards. Harley plans to arrange his cards in stacks so that each stack has the same number of cards. Also, each stack must have the same type of card (football or baseball). Every card in Harley's collection is to be placed in one of the stacks. What is the largest number of cards that can be placed in each stack?

K B

Numerade Educator

Problem 95

You and your brother both work the 4:00 P.M. to midnight shift. You have every sixth night off. Your brother has every tenth night off. Both of you were off on June 1. Your brother would like to see a movie with you. When will the two of you have the same night off again?

Nez Nikoo

Numerade Educator

Problem 96

A movie theater runs its films continuously. One movie is a short documentary that runs for 40 minutes. The other movie is a full-length feature that runs for 100 minutes. Each film is shown in a separate theater. Both movies begin at noon. When will the movies begin again at the same time?

Hira Hussain

Numerade Educator

Problem 97

Two people are jogging around a circular track in the same direction. One person can run completely around the track in 15 minutes. The second person takes 18 minutes. If they both start running in the same place at the same time, how long will it take them to be together at this place if they continue to run?

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 98

Two people are in a bicycle race around a circular track. One rider can race completely around the track in 40 seconds. The other rider takes 45 seconds. If they both begin the race at a designated starting point, how long will it take them to be together at this starting point again if they continue to race around the track?

Brandon Fox

Numerade Educator

Problem 99

If $a$ is a factor of $c$, what does this mean?

Cory Kuzinski

Numerade Educator

Problem 100

How do you know that 45 is divisible by 5 ?

Pammi Eswari

Numerade Educator

Problem 101

What does " $a$ is divisible by $b$ " mean?

Sarah Manchester

Sarah Manchester

Numerade Educator

Problem 102

Describe the difference between a prime number and a composite number.

Ebunoluwa Bolujo

Ebunoluwa Bolujo

Numerade Educator

Problem 103

What does the Fundamental Theorem of Arithmetic state?

Farnood Ensan

Numerade Educator

Problem 104

What is the greatest common divisor of two or more natural numbers?

Vysakh M

Numerade Educator

Problem 105

Describe how to find the greatest common divisor of two numbers.

Vysakh M

Numerade Educator

Problem 106

What is the least common multiple of two or more natural numbers?

Pagadala Kishore Reddy

Pagadala Kishore Reddy

Numerade Educator

Problem 107

Describe how to find the least common multiple of two natural numbers.

Debasish Das

Numerade Educator

Problem 108

The process of finding the greatest common divisor of two natural numbers is similar to finding the least common multiple of the numbers. Describe how the two processes differ.

Craig West

Numerade Educator

Problem 109

What does the Blitzer Bonus on page 256 have to do with Gödel's discovery about mathematics and logic, described on page 181 ?

Md.Daniyal Arshad

Md.Daniyal Arshad

Numerade Educator

Problem 110

In Exercises 110-113, determine whether each statement makes sense or does not make sense, and explain your reasoning.
I'm working with a prime number that intrigues me because it has three natural number factors.

Sheryl Ezze

Numerade Educator

Problem 111

Determine whether each statement makes sense or does not make sense, and explain your reasoning.
When I find the greatest common factor, I select common prime factors with the greatest exponent and when I find the least common multiple, I select common prime factors with the smallest exponent.

James Kiss

Numerade Educator

Problem 112

Determine whether each statement makes sense or does not make sense, and explain your reasoning.
I need to separate 70 men and 175 women into all-male or all-female teams with the same number of people on each team. By finding the least common multiple of 70 and 175 , I can determine the largest number of people that can be placed on a team.

Lauren Shelton

Numerade Educator

Problem 113

Determine whether each statement makes sense or does not make sense, and explain your reasoning.
(If you have not yet done so, read the Blitzer Bonus "GIMPS" on page 257.) I can find a prime number larger than the record prime $2^{74,207,281}-1$ by simply writing $2^{74,207,282}-1$.

Matthias Wuest

Numerade Educator

Problem 114

Write a four-digit natural number that is divisible by 4 and not by $8 .$

Kerry Thornton-Genova

Kerry Thornton-Genova

Numerade Educator

Problem 115

Find the greatest common divisor and the least common multiple of $2^{17} \cdot 3^{25} \cdot 5^{31}$ and $2^{14} \cdot 3^{37} \cdot 5^{30}$. Express answers in the same form as the numbers given.

Vysakh M

Numerade Educator

Problem 116

A middle-aged man observed that his present age was a prime number. He also noticed that the number of years in which his age would again be prime was equal to the number of years ago in which his age was prime. How old is the man?

Swati Agarwal

Numerade Educator

Problem 117

A movie theater runs its films continuously. One movie runs for 85 minutes and a second runs for 100 minutes. The theater has a 15 -minute intermission after each movie, at which point the movie is shown again. If both movies start at noon, when will the two movies start again at the same time?

Jingtian Li

Numerade Educator

Problem 118

The difference between consecutive prime numbers is always an even number, except for two particular prime numbers. What are those numbers?

Casey Evans

Numerade Educator

Problem 119

Wants to Be a Millionaire? How many total years during one's teen years is a person an age that's a prime number?
a. 1
b. 2
c. 3
d. 4

Hubert Agamasu

Numerade Educator

Problem 120

Use the divisibility rules listed in Table $5.1$ on page 253 to answer the questions in Exercises 120-122. Then, using a calculator, perform the actual division to determine whether your answer is correct.
Is $67,234,096$ divisible by 4 ?

Willis James

Numerade Educator

Problem 121

Use the divisibility rules listed in Table $5.1$ on page 253 to answer the questions. Then, using a calculator, perform the actual division to determine whether your answer is correct.
Is $12,541,750$ divisible by 3 ?

Sarah Wharton

Numerade Educator

Problem 122

Use the divisibility rules listed in Table $5.1$ on page 253 to answer the questions. Then, using a calculator, perform the actual division to determine whether your answer is correct.
Is $48,201,651$ divisible by 9 ?

Sarah Wharton

Numerade Educator

Problem 123

The following topics from number theory are appropriate for either individual or group research projects. A report should be given to the class on the researched topic. Useful references include liberal arts mathematics textbooks, books about numbers and number theory, books whose purpose is to excite the reader about mathematics, history of mathematics books, encyclopedias, and the Internet.
Euclid and Number Theory

AG

Ankit Gupta

Numerade Educator

Problem 124

The following topics from number theory are appropriate for either individual or group research projects. A report should be given to the class on the researched topic. Useful references include liberal arts mathematics textbooks, books about numbers and number theory, books whose purpose is to excite the reader about mathematics, history of mathematics books, encyclopedias, and the Internet.
An Unsolved Problem from Number Theory

AG

Ankit Gupta

Numerade Educator

Problem 125

The following topics from number theory are appropriate for either individual or group research projects. A report should be given to the class on the researched topic. Useful references include liberal arts mathematics textbooks, books about numbers and number theory, books whose purpose is to excite the reader about mathematics, history of mathematics books, encyclopedias, and the Internet.
Perfect Numbers

AG

Ankit Gupta

Numerade Educator

Problem 126

The following topics from number theory are appropriate for either individual or group research projects. A report should be given to the class on the researched topic. Useful references include liberal arts mathematics textbooks, books about numbers and number theory, books whose purpose is to excite the reader about mathematics, history of mathematics books, encyclopedias, and the Internet.
Deficient and Abundant Numbers

AG

Ankit Gupta

Numerade Educator

Problem 127

The following topics from number theory are appropriate for either individual or group research projects. A report should be given to the class on the researched topic. Useful references include liberal arts mathematics textbooks, books about numbers and number theory, books whose purpose is to excite the reader about mathematics, history of mathematics books, encyclopedias, and the Internet.
Formulas That Yield Primes

AG

Ankit Gupta

Numerade Educator

Problem 128

The following topics from number theory are appropriate for either individual or group research projects. A report should be given to the class on the researched topic. Useful references include liberal arts mathematics textbooks, books about numbers and number theory, books whose purpose is to excite the reader about mathematics, history of mathematics books, encyclopedias, and the Internet.
The Sieve of Eratosthenes

AG

Ankit Gupta

Numerade Educator