Math135Lecture09StudentNotes

# Math135Lecture09StudentNotes - Math 135 Lecture 9...

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Unformatted text preview: Math 135: Lecture 9: Consolidating What We’ve Learned Deﬁnition 1. An integer m divides an integer n, and we write m | n, if Proposition 2 (Transitivity of Divisibility TD). Let a, b, c ∈ Z. Proposition 3 (Divisibility of Integer Combinations DIC). Let a, b and c be integers. Proposition 4 (Bounds by Divisibility BBD). Deﬁnition 5 (Greatest Common Divisor). Let a and b be integers, 1. 2. Proposition 6 (GCD With Remainders GCD WR). Theorem 7 (GCD Characterization Theorem (GCD CT)). Extended Euclidean Algorithm (EEA) We conclude the EEA once we have rn+1 = 0. Then what equation can we get from row n? Suppose row n + 1 has Proposition 8 (Extended Euclidean Algorithm (EEA)). We can verify that rn | a and rn | b. We get a certiﬁcate that rn is the • The numbers from that satisfy together with • the Proposition 9 (Coprimeness and Divisibility (CAD)). Proposition 10 (GCD Of One (GCD OO)). Proposition 11 (Division by the GCD (DB GCD)). 1 Math 135: Lecture 9: Consolidating What We’ve Learned How do we show S = T ? 1. 2. How do we show S ⊆ T ? 1. 2. Theorem 12 (Linear Diophantine Equation Theorem Part 1 (LDET1) ). Theorem 13 (Linear Diophantine Equation Theorem Part 2 (LDET2)). Deﬁnition 14. Let S and T be two sets. A function f : S → T is onto (or surjective) if and only if 2 Math 135: Lecture 9: Consolidating What We’ve Learned Exercise 15. (From Fall 2009 Final) Prove that d = gcd(598 + 1, 599 + 3) = 2. 3 Math 135: Lecture 9: Consolidating What We’ve Learned Exercise 16. (From Fall 2009 Midterm) Prove that for all integers a, b and c, if a | c and b | c and gcd(a, b) = d then ab | cd. 4 Math 135: Lecture 9: Consolidating What We’ve Learned Exercise 17. (From Fall 2006 Final) Let d = gcd(3a + 2b, a + b). Prove that d = gcd(a, b). 5 ...
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Math135Lecture09StudentNotes - Math 135 Lecture 9...

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