Number Theory Fundamentals
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Questions and Answers

What is a divisor of a number?

  • An integer that divides it without leaving a remainder (correct)
  • An integer that multiplies it without leaving a remainder
  • An integer that subtracts it without leaving a remainder
  • An integer that divides it with a remainder

    • What is a prime number?

  • A positive integer less than 1 that is divisible by 2 and itself
  • A positive integer greater than 1 that is divisible by 1 and itself (correct)
  • A positive integer greater than 1 that is divisible by 2 and itself
  • A positive integer less than 1 that is divisible by 1 and itself

    • What is the purpose of the Euclidean Algorithm?

  • To compute the sum of two integers
  • To compute the GCD of two integers (correct)
  • To compute the difference of two integers
  • To compute the LCM of two integers

    • What does the symbol ≡ mean in a congruence?

    <p>Has the same remainder as</p> Signup and view all the answers

    What is a Diophantine equation?

    <p>A linear equation with at least one integer solution</p> Signup and view all the answers

    What does Fermat's Little Theorem state?

    <p>If p is a prime number, then a^p ≡ a (mod p) for any integer a</p> Signup and view all the answers

    What is the Fundamental Theorem of Arithmetic?

    <p>Every positive integer can be expressed as a product of prime numbers in a unique way</p> Signup and view all the answers

    What is a property of the GCD of two integers?

    <p>gcd(a, b) = gcd(b, a)</p> Signup and view all the answers

    Study Notes

    Divisibility

    • A divisor of a number is an integer that divides it without leaving a remainder.
    • Notation: a | b means "a divides b" or "a is a divisor of b".
    • Properties:
      • If a | b and a | c, then a | (b + c).
      • If a | b and a | c, then a | (b - c).
      • If a | b and c | a, then c | b.

    Prime Numbers

    • A prime number is a positive integer greater than 1 that is divisible only by 1 and itself.
    • Examples: 2, 3, 5, 7, 11, ...
    • Properties:
      • There are infinitely many prime numbers.
      • Every positive integer can be expressed as a product of prime numbers in a unique way ( Fundamental Theorem of Arithmetic).

    Greatest Common Divisor (GCD)

    • The GCD of two or more integers is the largest integer that divides each of them without leaving a remainder.
    • Notation: gcd(a, b) or (a, b).
    • Properties:
      • gcd(a, b) = gcd(b, a).
      • gcd(a, b) = gcd(a, b + a).
      • gcd(a, b) = gcd(a, b - a).

    Euclidean Algorithm

    • A method for computing the GCD of two integers.
    • Steps:
      1. Divide the larger number by the smaller number.
      2. Take the remainder as the new smaller number.
      3. Repeat steps 1 and 2 until the remainder is 0.
      4. The last non-zero remainder is the GCD.

    Congruences

    • A congruence is an equation of the form a ≡ b (mod n), where a, b, and n are integers.
    • Meaning: a and b have the same remainder when divided by n.
    • Properties:
      • a ≡ a (mod n).
      • If a ≡ b (mod n), then b ≡ a (mod n).
      • If a ≡ b (mod n) and b ≡ c (mod n), then a ≡ c (mod n).

    Diophantine Equations

    • A linear Diophantine equation is an equation of the form ax + by = c, where a, b, and c are integers.
    • Solution: x = x0 + (b/d)t, y = y0 - (a/d)t, where d = gcd(a, b) and x0, y0 are particular solutions.

    Fermat's Little Theorem

    • If p is a prime number, then a^p ≡ a (mod p) for any integer a.
    • Corollary: If p is a prime number, then a^(p-1) ≡ 1 (mod p) for any integer a not divisible by p.

    Divisibility

    • A divisor is an integer that divides another integer without leaving a remainder.
    • Notation: a | b means "a divides b" or "a is a divisor of b".
    • Properties of divisibility:
      • If a divides b and a divides c, then a divides b + c.
      • If a divides b and a divides c, then a divides b - c.
      • If a divides b and c divides a, then c divides b.

    Prime Numbers

    • A prime number is a positive integer greater than 1 that is divisible only by 1 and itself.
    • Examples: 2, 3, 5, 7, 11, ...
    • Properties of prime numbers:
      • There are infinitely many prime numbers.
      • Every positive integer can be expressed as a product of prime numbers in a unique way (Fundamental Theorem of Arithmetic).

    Greatest Common Divisor (GCD)

    • The GCD of two or more integers is the largest integer that divides each of them without leaving a remainder.
    • Notation: gcd(a, b) or (a, b).
    • Properties of GCD:
      • gcd(a, b) = gcd(b, a).
      • gcd(a, b) = gcd(a, b + a).
      • gcd(a, b) = gcd(a, b - a).

    Euclidean Algorithm

    • A method for computing the GCD of two integers using repeated divisions.
    • Steps:
      • Divide the larger number by the smaller number.
      • Take the remainder as the new smaller number.
      • Repeat steps until the remainder is 0.
      • The last non-zero remainder is the GCD.

    Congruences

    • A congruence is an equation of the form a ≡ b (mod n), where a, b, and n are integers.
    • Meaning: a and b have the same remainder when divided by n.
    • Properties of congruences:
      • a ≡ a (mod n).
      • If a ≡ b (mod n), then b ≡ a (mod n).
      • If a ≡ b (mod n) and b ≡ c (mod n), then a ≡ c (mod n).

    Diophantine Equations

    • A linear Diophantine equation is an equation of the form ax + by = c, where a, b, and c are integers.
    • Solution: x = x0 + (b/d)t, y = y0 - (a/d)t, where d = gcd(a, b) and x0, y0 are particular solutions.

    Fermat's Little Theorem

    • If p is a prime number, then a^p ≡ a (mod p) for any integer a.
    • Corollary: If p is a prime number, then a^(p-1) ≡ 1 (mod p) for any integer a not divisible by p.

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    Explore the basics of number theory, including divisors, prime numbers, and their properties. Learn to identify divisors, understand prime numbers, and discover their characteristics.

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