Lecture 2.ppt - Basic Number Theory Agenda Divisors Primality Fundamental Theorem of Arithmetic Division Algorithm Greatest common divisors/least common

Lecture 2.ppt - Basic Number Theory Agenda Divisors...

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Basic Number Theory
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2 Agenda Divisors Primality Fundamental Theorem of Arithmetic Division Algorithm Greatest common divisors/least common multiples Relative Primality Modular arithmetic Caesar s Cipher
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L9 3 Importance of Number Theory Before the dawn of computers, many viewed number theory as last bastion of pure math which could not be useful and must be enjoyed only for its aesthetic beauty. No longer the case. Number theory is crucial for encryption algorithms. Of utmost importance to everyone from Bill Gates, to the CIA E.G., of great importance Network Security .
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L9 4 Importance of Number Theory The encryption algorithms depend heavily on modular arithmetic. We need to develop various machinery (notations and techniques) for manipulating numbers before can describe algorithms in a natural fashion. First we start with divisors.
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L9 5 Divisors DEF: Let a , b and c be integers such that a = b · c . Then b and c are said to divide (or are factors ) of a, while a is said to be a multiple of b (as well as of c ). The pipe symbol | denotes divides so the situation is summarized by: b | a c | a . NOTE: Students find notation confusing, and think of | in the reverse fashion, perhaps confuse pipe with forward slash /
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L9 6 Divisors. Examples Q: Which of the following is true? 1. 77 | 7 2. 7 | 77 3. 24 | 24 4. 0 | 24 5. 24 | 0
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L9 7 Divisors. Examples A: 1. 77 | 7: false bigger number can t divide smaller positive number 2. 7 | 77: true because 77 = 7 · 11 3. 24 | 24: true because 24 = 24 · 1 4. 0 | 24: false, only 0 is divisible by 0 5. 24 | 0: true, 0 is divisible by every number (0 = 24 · 0)
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8 Formula for Number of Multiples up to given n Q: How many positive multiples of 15 are less than 100?
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L9 9 Formula for Number of Multiples up to given n A: Just list them: 15, 30, 45, 60, 75, 90. Therefore the answer is 6. Q: How many positive multiples of 15 are less than 1,000,000?
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L9 10 Formula for Number of Multiples up to Given n A: Listing is too much of a hassle. Since 1 out of 15 numbers is a multiple of 15, if 1,000,000 were divisible by 15, answer would be exactly 1,000,000/15. However, since 1,000,000 isn t divisible by 15, need to round down to the highest multiple of 15 less than 1,000,000 so answer is 1,000,000/15 . In general: The number of d -multiples less than N is given by: |{ m Z + | d | m and m N }| = N / d
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L9 11 Divisor Theorem THM: Let a, b, and c be integers. Then: 1. a | b a | c a |( b + c ) 2. a | b a | bc 3. a | b b | c a | c EG: 1. 17|34 17|170 17|204 2. 17|34 17|340 3. 6|12 12|144 6 | 144
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L9 12 Divisor Theorem. Proof of no. 2 In general, such statements are proved by starting from the definitions and manipulating to get the desired results. EG. Proof of no. 2 ( a | b a | bc ): Suppose a | b. By definition, there is a number m such that b = am . Multiply both sides by c to get bc = amc = a ( mc ). Consequently, bc has been expressed as a times the integer mc so by definition of | , a | bc
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L9 13 Prime Numbers DEF: A number n 2 prime if it is only divisible by 1 and itself. A number n 2 which isn t prime is called composite .
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