Number Theory – Study Guide for Test 1
Divisibility in the Integers
2.1 Division Algorithm
Forms of integers (for example, 4k, 4k+1, 4k+2, 4k+3)
Definition of divides and properties
2.2 Definition of greatest common divisor
Theorem 2.3, 2.4 and their Corollaries
Euclid’s Lemma
2.3 Euclidean Algorithm
Write gcd(a, b) = ax + by
2.4 Linear Diophantine Equations (Theorem 2.9)
Prime Numbers
3.1 Definition of prime and composite; Fundamental Theorem of Arithmetic
3.2 Sieve of Eratosthenes; Euclid’s proof that there are infinitely many primes
3.3 Gaps in primes
Congruences
4.2 Definiton of a ≡ b (mod n)
Properties (Theorem 4.2); Cancellation (Theorem 4.3)
4.3 Divisibility tests; Congruence (mod 9) and (mod 11)
4.4 Linear congruences (Theorem 4.7); Chinese Remainder Theorem (Theorem 4.8)
5.3 Fermat’s Little Theorem
Know How To
1. State definitions
2. State theorems by name:
Division Algorithm, Euclid’s Lemma, Fundamental Theorem of Arithmetic,
Chinese Remainder Theorem, Fermat’s Little Theorem
3. Use propositions, theorems, and their corollaries (for computations and short proofs)
4. Prove:
Theorem 2.3 (using the Well-Ordering Principle)
There are infinitely many primes (Euclid’s proof)
Fermat’s Little Theorem
Study class notes, collected homework, and other homework
The other homework was:
p19 #2, 3ab, 4, 8, 11; p25 #2d, 3, 6b, 14a, 20abe; p31 #1, 2ad; p38 #1, 2a, 3c, 6a, 9bc;
p44 #3ae, 4, 7, 10, 12, 16; p50 #1, 2, 5; p59 #1, 9a; p68 #1ab, 2, 4, 6b, 8a, 16Part1;
p73 #3, 5, 8, 10, 22; p82 #1abcd, 2a, 4a, 5, 6, 18; p96 #2ac,4b,6,7