# Number Theory

**Introduction**

Number theory is an exciting area of mathematics, with many practical applications. Some day, you may need to use factors to help you build a yard with a specific area, or use multiples to solve a scheduling problem at work. Many topics from elementary number theory are covered on Math Goodies. You have been selected to explore these topics, and to apply your knowledge through critical thinking.

**The Task**

You will explore resources on Math Goodies to answer specific questions on number theory. The key to learning in this WebQuest is to become engaged in finding information, and to learn from what you have found. These tasks will require a computer, access to the Internet, paper and pencil. Familiarity with Math Goodies is helpful, but not necessary.

**The Process**

**Process Part I: Number Theory Unit and Worksheets**

- Browse our instructional unit on Elementary Math (Number Theory), which covers factors, GCF, Multiples, LCM, prime and composite numbers, divisibility tests and exponents.
- Complete all five exercises at the end of each core lesson in this unit.
- Complete our Worksheet on The Sieve of Eratosthenes

**Elementary Number Theory Questions**

Answer each question, in order, and write down your answer on paper in your own words. You may work alone or in groups.

- What is elementary number theory?
- Give an example of how can multiples help you with gardening? (You may provide a specific example you found on our site.)
- What is a factor?
- What is 1,729,463 raised to the zero power?
- Is 2
^{5}the same as 2 x 5? Explain why or why not using full sentences. - What is the Fibonacci sequence? (Hint: It can be found on several pages.)
- Write the Fibonacci sequence on paper.
- What is the rule for this sequence? Write your answer using your own words.
- Is the number 31 prime or composite? Explain your answer.
- Is the number 747 prime or composite? Explain your answer.
- How did you arrive at your answer to questions 9 and question 10? What method did you use for each?
- Why do we use divisibility tests?
- Which test(s) will determine if the number 876 is prime or composite? Explain your answer.
- What is the Sieve of Eratosthenes?
- Find all prime numbers less than 100.
- What is the smallest prime number?
- Which prime number is even?
- What is an emirp?
- List all emirps between 1 and 100.

**Resources**

- Our instructional unit on Elementary Math.
- Our worksheet on The Sieve of Eratosthenes.

**Self-Evaluation**

- Switch your answers with a classmate or group member.
- Review each other’s answers.
- Exchange ideas: Discuss what you have learned.
- Go through the answer key to our worksheet on The Sieve of Eratosthenes.
- Assess your ability to find information in this WebQuest.
- Assess your ability to learn from the information you found.
- Which tasks were easy for you?
- Which tasks did you struggle with?

**Conclusion**

Congratulations! You learned about topics in number theory and extended your knowledge through critical thinking. You did this by embarking on a quest for information, and by learning from the information you found.

Number theory is a branch of mathematics that focuses on the study of whole numbers, or integers, and the relationships between them. It explores patterns, properties, and behaviors of numbers, often in their purest form without considering their application to real-world problems. Number theory delves into various topics, including prime numbers, divisibility, congruences, and arithmetic functions.

One of the central concepts in number theory is prime numbers. These are numbers greater than 1 that can only be divided by 1 and themselves without leaving a remainder. Understanding prime numbers is crucial in cryptography, a field that deals with secure communication systems.

Number theory also deals with divisibility rules and the relationships between numbers when they are divided. For instance, it explores concepts like greatest common divisors and least common multiples.