Test 1 Study Guide

The tiny syllabus for Test 1 is based on Secs. 1.1,1.2,1.3,1.4 in the textbook (Rotman 2ed). Explicitly, you should be able to
  1. State precisely all definitions of bold-face terms in these sections.
  2. State precisely the Least Integer Axiom, the Least Criminal Axiom, and Theorems 1.4 and 1.10.
  3. Discuss the logical relationship among the 4 statements in 2)
  4. Produce brief inductive proofs as in the exercises.
  5. Produce brief proofs using one of the Axioms as in Prop. 1.1, Thm. 1.2.
  6. State the 3-term Pascal Identity (Lemma 1.15) and be able to use it to verify facts about binomial coefficients.
  7. State the Binomial Theorem (Corollary 1.18) and be able to use it as in exercises assigned.
  8. Know the Division Algorithm and be able to state it as a Theorem.
  9. Prove the infinitude of the set of primes.
  10. State Euclid's Lemma, and its converse (cf Exercise 1.40)
  11. State the representability of gcd(a,b) (Thm. 1.29)
  12. Use the "Euclidean Algorithm" for gcd both to find gcd and represent it.
  13. State precisely the Fundamental Theorem of Arithmetic

Note: Since FTA comes last in this syllabus, it may not be assumed in proofs unless the question specifically authorizes it.

Len Smiley