Math 430 Review Chapter 0 Put Your Name In Here For this assignment, you should copy this document as closely as possible. Put your name in the Summary: Math 430 Review Chapter 0 Put Your Name In Here For this assignment, you should copy this document as closely as possible. Put your name in the top right corner. These are some concepts from Chapter 0 that you should already know. 1. Definition (The Well Ordering Principle) - Every nonempty set of positive integers con- tains a smallest member. 2. Theorem (The Division Algorithm) - Let a and b be integers with b > 0. Then there exist unique integers q and r with the property that a = bq + r, where 0 r < b. 3. Definition - The Greatest Common Divisor of two nonzero integers a and b is the largest of all common divisors of a and b. We denote this integer by gcd(a, b). When gcd(a, b) = 1, we say a and b are relatively prime. 4. Theorem For any nonzero integers a and b, there exist integers s and t such that gcd(a, b) = as + bt. Moreover, gcd(a, b) is the smallest positive integer of the form as + bt. 5. Corollary If a and b are relatively prime, then there exist integers s and t such that as+bt = 1. 6. Theorem (Euclid's Lemma) If p is a prime that divides ab, then p divides a or p divides b (or both). Proof: Suppose that p is a prime that divides ab, but without loss of generality (WLOG) does not divide a. Then we must show that p divides b. Since p does not divide a, then a and p are relatively prime. So there exist integers s and t such that 1 = as + pt. Multiply through by b to get b = abs + ptb. Since p divides ab and p divides a, p divides the right hand side of the equation. Hence p divides the left as well. So p divides b. . Collections: Mathematics