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Unformatted text preview: CS103A HO#34 More Number Theory 2/13/08 1 CS103A 2/13/08 If you want a text: Kenneth Rosen Discrete Mathematics and Its Applications, 6 th Ed. Some properties of gcd If a = bq + r for integers a, b, q, and r, then gcb(a, b) = gcd(b, r) . gcd(66, 45) = gcd(45, 21) Theorem 30.4 (proved in the handout) Some properties of gcd The GCD Identity (Bzout's Identity) If gcd(a, b) = d , then there exist integers m and n such that d = am + bn . tienne Bzout 17301783 French mathematician who is best known for his theorem on the number of solutions of polynomial equations. Some properties of gcd The GCD Identity (Bzout's Identity) If gcd(a, b) = d , then there exist integers m and n such that d = am + bn . The gcd of a and b can be expressed as a linear combination of a and b. Some properties of gcd The GCD Identity (Bzout's Identity) If gcd(a, b) = d , then there exist integers m and n such that d = am + bn . 1082 3 422 2242 1902156212222228823 633 333 2693 11353201312673233333 m n d a = 66, b = 45 Some properties of gcd The GCD Identity (Bzout's Identity) If gcd(a, b) = d , then there exist integers m and n such that d = am + bn . m n a = 66, b = 451 66+2 45 333 288 243 198 153 108 63 3 267 222 177 132 87 423 2 201 156 111 66 212469 1 135 90 454590135 69 2421661111562011 342871321772222672631081531982432883333 3 2 1123 CS103A HO#34 More Number Theory 2/13/08 2 Some properties of gcd The GCD Identity (Bzout's Identity) If gcd(a, b) = d , then there exist integers m and n such that d = am + bn . m n a = 66, b = 452 66 + 3 45 gcd(66, 45) = = 3 333 288 243 198 153 108 63 3 267 222 177 132 87 423 2 201 156 111 66 212469 1 135 90 454590135 69 2421661111562011 342871321772222672631081531982432883333 3 2 1123 If gcd(a, b) = d , then there exist integers m and n such that d = am + bn . (a and b are not both 0.) PROOF: Let S be the set of all positive integers of the form am + bn , and let k = au + bv be the smallest member of S. (We need to argue that S is not empty and that it has a smallest member). If gcd(a, b) = d , then there exist integers m and n such that d = am + bn . (a and b are not both 0.) PROOF: Let S be the set of all positive integers of the form am + bn, and let k = au + bv be the smallest member of S. (We need to argue that S is not empty and that it has a smallest member). First we show that k  a and k  b. If a = 0, then k  a. If gcd(a, b) = d , then there exist integers m and n such that d = am + bn . (a and b are not both 0.) PROOF: Let S be the set of all positive integers of the form am + bn, and let k = au + bv be the smallest member of S. (We need to argue that S is not empty and that it has a smallest member)....
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This note was uploaded on 10/01/2011 for the course CS 103A taught by Professor Plummer,r during the Winter '07 term at Stanford.
 Winter '07
 Plummer,R
 Computer Science

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