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
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. .