34 Slides--More Number theory

34 Slides--More Number theory - CS103A HO#34 More Number...

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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 1730-1783 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 . 108-2 3 42-2 2-24-2 1-90-2-156-2-1-222-2-2-288-2-3 63-3 3-3-3 2-69-3 1-135-3-201-3-1-267-3-2-333-3-3 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 = 45-1 66+2 45 333 288 243 198 153 108 63 3 267 222 177 132 87 42-3 2 201 156 111 66 21-24-69 1 135 90 45-45-90-135 69 24-21-66-111-156-201-1 3-42-87-132-177-222-267-2-63-108-153-198-243-288-333-3 3 2 1-1-2-3 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 = 45-2 66 + 3 45 gcd(66, 45) = = 3 333 288 243 198 153 108 63 3 267 222 177 132 87 42-3 2 201 156 111 66 21-24-69 1 135 90 45-45-90-135 69 24-21-66-111-156-201-1 3-42-87-132-177-222-267-2-63-108-153-198-243-288-333-3 3 2 1-1-2-3 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.

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34 Slides--More Number theory - CS103A HO#34 More Number...

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