Lesson 10: Congruence, Euclid

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Lesson 10 Congruence, Euclid

• Objectives
• Determine if two integers are congruent, modulo a
  third
• Calculate simple hashing functions
• Generate pseudorandom numbers
• Use the Euclidean algorithm to find the GCD

• Outline
• Congruence
• Hashing
• Pseudorandom Number Generation
• Euclidean Algorithm
• HW
• 2.5 Assignment from Lesson 9
• 2.6 2beh, 6, 7, 12, 18, 19, 25, 36bdf

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Section 2.5

• Binary Expansions
• Base Representations
• Base Arithmetic
• Ones, Twos complement

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Base Problem

• Find a,b,c
• (abc)5 (cba)8

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Euclidean Algorithm

• Euclid (circa 325-265 B.C.)
• Divide the larger of the two integers by the
  smaller
• Divide the divisor by the remainder
• Continue until there is no remainder
• The last remainder is the GCD

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Euclidean Algorithm

• GCD ( 325, 265 )
• GCD ( 432, 459 )
• GCD ( 175, 130 )
• GCD ( 287, 91 )