lecnotes2 - Lecture Notes, Concepts of Mathematics (21-127)...

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Lecture Notes, Concepts of Mathematics (21-127) Lecture 1, Recitation A–D, Spring 2008 2 Integers and Diophantine Equations 2.1 The Division Algorithm READ: Textbook pages 25–34. EXERCISES: Pages 50–55, #1–30, 73, 74 Recall the definition of divisibility: If a and b are integers, then a divides b iff there is an integer q such that aq = b . Proposition 2.1 (Proposition 2.11, page 25) Let a , b , and c be integers. (i) If a | b and b | c , then a | c . (ii) If a | b and a | c , then a | ( bx + cy ) for any x, y Z . In particular, a | ( b + c ) and a | ( b - c ) . (iii) If a | b and b | a , then a = ± b . (iv) If a | b and b 6 = 0 , then | a | ≤ | b | . Proof: (ii) Assume a | b , a | c , and let x, y Z . Since a | b and a | c , there are q 1 , q 2 Z such that aq 1 = b and aq 2 = c . Then we have aq 1 x = bx and aq 2 y = cy , and thus bx + cy = aq 1 x + aq 2 y = a ( q 1 x + q 2 y ) . Since
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This note was uploaded on 01/25/2010 for the course MATH 21127 taught by Professor Howard during the Spring '08 term at Carnegie Mellon.

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lecnotes2 - Lecture Notes, Concepts of Mathematics (21-127)...

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