Divisibility and Prime Factorization Quiz

Divisibility

• Divisibility is a mathematical concept referring to integers.
• If a, b are integers and a ≠ 0, then 'a divides b' (denoted as a|b) if there's an integer m such that b = am.
• A natural number p (not equal to 1) is prime if its only positive factors are p and 1.
• Any number greater than 1 that is not prime is a composite number.

Prime Factorization

• Every natural number greater than 1 has a unique prime factorization.
• The prime factorization of a natural number expresses it as a product of prime numbers raised to unique powers.
• Example: 2400 = 2⁵ × 3¹ × 5².

Relatively Prime Integers

• Two integers a and b are relatively prime if their greatest common divisor (gcd) is 1.
  • This means they share no common factors greater than 1.

Euclidean Algorithm

• A method for finding the greatest common divisor (gcd) of two integers.
• It involves repeatedly applying the division algorithm.
• The gcd of two integers is expressed as an integer combination of the two integers.

Diophantine Equations

• A linear equation of the form ax + by = c, where a, b, and c are integers.
• A solution is an ordered pair of integers (x, y) that satisfy the equation.
• If gcd(a, b)|c, then there are infinitely many integer solutions.
• If gcd(a, b)|c, then there is no integer solution.
• If a solution is found (x₁, y₁), then all other solutions are of the form: x = x₁ + b/d * k and y = y₁ − a/d * k, where d = gcd(a,b), k ∈ Z.

Elementary Proof Techniques

• Direct Proof: Assume the hypothesis is true and deduce the conclusion.
• Contrapositive Proof: Assuming the negation of the conclusion and showing that leads to the negation of the hypothesis.
• Contradiction Proof (Indirect Proof): Assume the negation of the conclusion is true and show this leads to a contradiction of the hypothesis and/or previously established fact.
• Proof by Induction: Proving a statement for all natural numbers (e.g., 1,2,3...). It involves a basis step (proving the statement is true for a starting natural number) and an inductive step (proving if the statement is true for a natural number k, it is also true for k+1).

The Dart Board Problem

• The problem concerns the set of positive integers that can be generated by combining a and b.
• When a and b are relatively prime, the set of achievable values is (ab − a − b, ∞).

Test your knowledge on divisibility, prime numbers, and the Euclidean algorithm. This quiz covers concepts such as prime factorization, relatively prime integers, and identifies methods to find the greatest common divisor. Perfect for mathematics enthusiasts and students alike!