CS 173: Skills list for Examlet 2

Proof Techniques

• Write a simple direct proof, using familiar concepts, with good mathematical style. Make sure your statements are in logical order, starting with the given information and ending with what you needed to show.
• Write a proof by cases
• Convert a claim to its contrapositive and prove that using direct proof.
• Know the following standard ways to approach parts of a proof:
  • prove a universal claim by working with a 'representative' object of the appropriate type (not a concrete example)
  • prove an if/then statement by assuming whatever's in the hypothesis and proving the conclusion
  • disprove a universal claim by giving a concrete counterexample
  • prove an existential claim by giving specific values that make the claim true

Number Theory

• Know definitions, decide if simple statements involving them are true, understand the corresponding shorthand notation:
  • a divides b, b is a multiple of a,
  • x is odd, x is even
  • x is prime, x is composite
  • x is a perfect square
  • x is a rational number
  • x and y are relatively prime
• Apply the definition of divides to zero and to negative numbers (these aren't really 'edge cases' - the same exact definition applies!). For example, explain why zero is an even number.
• Prime numbers, factoring a number into primes. (Where we only consider numbers >= 2.)
• Special cases for prime: 0 and 1 are not prime (primes must be greater than or equal to 2).
• Know the definitions of gcd(a,b) and lcm(a,b) and be able to compute gcd(a,b) and lcm(a,b) for specific (smallish) values of a and b.
• Write any (small) positive integer as a product of primes.
• Know that sqrt(2) is not rational.
• Know that there are infinitely many primes.
• Be able to state the Fundamental Theorem of Arithmetic: Any integer >= 2 can be written as the product of primes and each integer has only one prime factorization (except for the order in which you write the factors).
• Define what it means for x and y to be congruent mod k
• Be able to state the "Division Algorithm" theorem.
• Compute the gcd of two larger numbers using the Euclidean algorithm, i.e. trace the execution of the algorithm.
• Prove simple number theory claims using direct proof and the definitions of the terms involved. For example, if a divides b and a divides c, then a divides b+c.