Free Printable Worksheets for learning Number theory at the College level

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Number Theory

Number theory is a branch of mathematics that studies whole numbers, including prime numbers, composite numbers, and their properties.

Key Concepts

  • Prime Numbers: Whole numbers that are divisible only by themselves and 1, such as 2, 3, 5, 7, 11.
  • Composite Numbers: Whole numbers that are divisible by more than two numbers, such as 4, 6, 8, 9, 10.
  • Divisibility: A whole number is divisible by another whole number if it can be divided evenly without a remainder. For example, 12 is divisible by 3 because 12/3 = 4.
  • GCD and LCM: GCD (Greatest Common Divisor) is the largest whole number that divides two or more whole numbers without leaving a remainder. LCM (Least Common Multiple) is the smallest whole number that is a multiple of two or more whole numbers.
  • Modular Arithmetic: A system of arithmetic that calculates the remainder when a number is divided by another number. It is often used in cryptography and computer science.

Important Information

  • The Fundamental Theorem of Arithmetic: Every composite number can be uniquely expressed as a product of primes.
  • Number theory has applications in cryptography, computer science, and many other fields.
  • Number theory is an active area of research and has many unsolved problems, such as the Riemann Hypothesis.

Takeaways

  • Number theory explores the properties of whole numbers, including prime and composite numbers.
  • Divisibility, GCD, LCM, and modular arithmetic are important concepts in number theory.
  • The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes.
  • Number theory has many practical applications and is an active area of research with many unsolved problems.

Here's some sample Number theory vocabulary lists Sign in to generate your own vocabulary list worksheet.

Word Definition
Integer A whole number that is not a fraction or decimal. Example: 0, 1, 2, 3, 4
Prime A number greater than 1 that can only be divided by 1 and itself. Example: 2, 3, 5, 7, 11
Composite A number that can be divided by more than just 1 and itself. Example: 4, 6, 8, 9, 10
Factor A number that divides exactly into another number. Example: 3 is a factor of 6 because 6 ÷ 3 = 2
Divisor A number that divides another number exactly. Example: 4 is a divisor of 12 because 12 ÷ 4 = 3
Multiple The product of a number and an integer. Example: 20, 40, 60 are multiples of 10
Greatest Common Factor (GCF) The highest common factor that divides exactly into two or more numbers. Example: GCF of 12 and 18 is 6
Least Common Multiple (LCM) The smallest multiple that two or more numbers have in common. Example: LCM of 4 and 6 is 12
Even A number that is divisible by 2. Example: 4, 8, 12, 16
Odd A number that is not divisible by 2. Example: 3, 5, 7, 9
Divisibility The ability for one number to be divided exactly by another number. Example: 12 is divisible by 3 because 12 ÷ 3 = 4
Decimal A number expressed using a decimal point to represent fractional parts. Example: 0.5, 1.25, 3.333
Recurring Decimal A decimal that repeats indefinitely without terminating. Example: 0.666... which is the same as 0.6 overline
Fraction A number that represents a part of a whole or group. Example: 1/2, 2/3, 3/4
Improper Fraction A fraction with a numerator that is greater than or equal to its denominator. Example: 7/4, 5/3, 9/2
Mixed Number A whole number combined with a fraction. Example: 2 1/3, 3 1/2, 4 3/4
Rational Number A number that can be expressed as a fraction where the numerator and denominator are both integers. Example: 1/2, 3/4, 5/8
Irrational Number A number that cannot be expressed as a fraction where the numerator and denominator are both integers. Example: √2, √3, π
Real Number A number that can be expressed as a decimal or a fraction, including rational and irrational numbers. Example: 1, 0.5, -3, √7
Imaginary Number A number that is defined as a multiple of the square root of -1. Example: 2i, 3i, -5i

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Number theory

Number theory is a branch of mathematics that deals with the properties and behavior of numbers, particularly integers.

Prime Numbers

Prime numbers are integers that are divisible only by 1 and themselves. Some important facts about prime numbers include:

  • 1 is not a prime number.
  • There are infinitely many prime numbers.
  • Prime numbers have many applications in cryptography and coding theory.

Divisibility

Divisibility is a fundamental concept in number theory. An integer a is said to be divisible by another integer b if a divided by b yields an integer without any remainder. Some important results related to divisibility include:

  • If a is divisible by b, and b is divisible by c, then a is divisible by c.
  • If a is divisible by b and c, then it is also divisible by the least common multiple of b and c.
  • If a is divisible by p, where p is prime, and p does not divide b, then a does not divide b.

Modular Arithmetic

Modular arithmetic deals with the properties of integers that remain unchanged when divided by a fixed integer. Some key topics within modular arithmetic include:

  • Modular addition and subtraction
  • Modular multiplicative inverses
  • Modular exponentiation

Diophantine Equations

Diophantine equations are equations in which the solutions must be integers. Solving Diophantine equations is a major focus of number theory, and some examples include:

  • Fermat's Last Theorem
  • Pythagorean Triples

Continued Fractions

A continued fraction is a representation of a real number as an infinite sequence of ratios of integers. Some important results in continued fractions include:

  • Every irrational number can be represented as a continued fraction.
  • The most well-known continued fraction is the golden ratio, which has a representation of [1; 1, 1, 1, ...].

By understanding these key concepts in number theory, you will be able to deepen your understanding of a wide range of mathematical fields.

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Number Theory Practice Sheet

Solve the following problems:

  1. Find the GCD and LCM of 36 and 48.
  2. Prove that there exist infinitely many prime numbers.
  3. Define prime factorization and give an example of the prime factorization of a number.
  4. What is the Euler totient function and how is it calculated?
  5. Find the last digit of $7{2020}$.
  6. Define modular arithmetic and solve the following congruence equation: $5x \equiv 1 \pmod{7}$.
  7. Use the Chinese Remainder Theorem to solve the following system of congruence equations: $$ \begin{cases} x \equiv 2 \pmod{3}\ x \equiv 3 \pmod{5} \end{cases} $$
  8. Define perfect numbers and find the first four perfect numbers.
  9. State Fermat's Little Theorem and use it to find $37{45} \pmod{11}$.
  10. Define Mersenne primes and find the first three Mersenne primes.

Bonuses: 11. Prove that every even integer greater than 2 can be expressed as the sum of two prime numbers. 12. Prove that there are no solutions to $x2 - 3y2 = 5$ in integers $x$ and $y$. 13. Define twin primes and find the first four twin primes.

Number Theory Practice Sheet

Prime Numbers

  1. What is a prime number?
  2. What is the definition of a prime number?
  3. What is the difference between a prime number and a composite number?
  4. How do you determine if a number is prime or composite?
  5. What is the smallest prime number?
  6. What is the largest prime number?
  7. What is the formula for finding prime numbers?
  8. What is the Sieve of Eratosthenes?
  9. How can the Sieve of Eratosthenes be used to find prime numbers?
  10. What is a prime factor?

Divisibility

  1. What is the definition of divisibility?
  2. What are the divisibility rules?
  3. How can divisibility rules be used to determine if a number is divisible by another number?
  4. What is the divisibility test?
  5. How can the divisibility test be used to determine if a number is divisible by another number?
  6. What is the greatest common factor (GCF)?
  7. How can the GCF be used to determine if a number is divisible by another number?
  8. What is the least common multiple (LCM)?
  9. How can the LCM be used to determine if a number is divisible by another number?

Number Theory

  1. What is number theory?
  2. What are the applications of number theory?
  3. What is the law of quadratic reciprocity?
  4. What is the Fermat's Little Theorem?
  5. How can Fermat's Little Theorem be used to determine if a number is prime?
  6. What is the Chinese Remainder Theorem?
  7. How can the Chinese Remainder Theorem be used to solve number theory problems?
  8. What is the Euclidean Algorithm?
  9. How can the Euclidean Algorithm be used to solve number theory problems?
  10. What is the Pythagorean Theorem?

Number Theory Practice Sheet

Question 1

Given two integers a and b, find the greatest common divisor (gcd) of a and b.

Question 2

Let n be a positive integer. Prove that n is divisible by 3 if and only if the sum of its digits is divisible by 3.

Question 3

Prove that the product of two prime numbers is also a prime number.

Question 4

Prove that for any two integers a and b, the greatest common divisor (gcd) of a and b is the same as the gcd of b and a mod b.

Question 5

Let n be a positive integer. Prove that n is divisible by 4 if and only if the last two digits of n are divisible by 4.

Question 6

Let n be a positive integer. Prove that n is divisible by 5 if and only if the last digit of n is 0 or 5.

Question 7

Prove that the product of two even numbers is always even.

Question 8

Let n be a positive integer. Prove that n is divisible by 6 if and only if it is divisible by both 2 and 3.

Question 9

Let n be a positive integer. Prove that n is divisible by 8 if and only if the last three digits of n are divisible by 8.

Question 10

Let n be a positive integer. Prove that n is divisible by 9 if and only if the sum of its digits is divisible by 9.

Here's some sample Number theory quizzes Sign in to generate your own quiz worksheet.

Problem Answer
What is a prime number? A number that is only divisible by 1 and itself
What is the Fundamental Theorem of Arithmetic? Every positive integer can be uniquely factored into a product of primes
What is the Euclidean Algorithm used for? To find the greatest common divisor of two integers
State Fermat's Little Theorem. If p is a prime number and a is an integer, then ap - a is divisible by p
What is quadratic reciprocity used for? To determine if there exists an integer solution to a quadratic equation modulo p
Give the statement of the Goldbach Conjecture. Every even integer greater than 2 can be expressed as the sum of two primes
What is modular arithmetic used for? To perform operations on the remainders of integers after division by a fixed integer
State the Chinese Remainder Theorem. Given integers n1, n2, ..., nk that are pairwise coprime, and given any integers a1, a2, ..., ak, there exists an integer x solving the system of simultaneous congruences, and any two solutions of the congruence system are congruent modulo n1 * n2 * ... *n_k
What is a Mersenne prime? A prime number of the form 2p - 1, where p is also prime
What is the Riemann Hypothesis? Each nontrivial zero of the Riemann zeta function has a real part equal to 1/2
Question Answer
What is the definition of a prime number? A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
What is the Fundamental Theorem of Arithmetic? The Fundamental Theorem of Arithmetic states that every positive integer can be written as a product of prime numbers in a unique way, up to the order of the factors.
What is the definition of a composite number? A composite number is a natural number greater than 1 that is not prime.
What is the definition of a perfect number? A perfect number is a positive integer that is equal to the sum of its proper divisors.
What is the definition of an abundant number? An abundant number is a positive integer for which the sum of its proper divisors is greater than the number itself.
What is the definition of a deficient number? A deficient number is a positive integer for which the sum of its proper divisors is less than the number itself.
What is the definition of a square number? A square number is an integer that can be expressed as the product of two equal integers.
What is the definition of a cube number? A cube number is an integer that can be expressed as the product of three equal integers.
What is the definition of a triangular number? A triangular number is an integer that can be expressed as the sum of consecutive positive integers, starting with 1.
Question Answer
What is the definition of a prime number? A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
What is the Fundamental Theorem of Arithmetic? The Fundamental Theorem of Arithmetic states that every positive integer can be written as a product of prime numbers in a unique way, up to the order of the factors.
What is the definition of a composite number? A composite number is a natural number greater than 1 that is not prime.
What is the definition of a perfect number? A perfect number is a positive integer that is equal to the sum of its proper divisors, which are all the divisors of the number other than itself.
What is the definition of an abundant number? An abundant number is a positive integer for which the sum of its proper divisors is greater than the number itself.
What is the definition of a deficient number? A deficient number is a positive integer for which the sum of its proper divisors is less than the number itself.
What is the definition of a Euclidean algorithm? The Euclidean algorithm is an algorithm used to find the greatest common divisor of two integers.
What is the definition of a modulo operation? A modulo operation is an operation that returns the remainder of a division between two numbers.
What is the definition of a congruence relation? A congruence relation is an equivalence relation between two integers, a and b, which states that a and b are congruent modulo n, where n is a positive integer.
What is the definition of a modular inverse? A modular inverse is an integer x such that (x * a) mod n = 1, where a and n are two integers.
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