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Sets, Combinatorics, Probability, and Number Theory Mathematical Structures for Computer Science Chapter 3 Copyright © 2006 W.H. Freeman & Co.MSCS Slides Probability

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Section 3.7Number Theory1 Fundamental Theorem of Arithmetic FUNDAMENTAL THEOREM OF ARITHMETIC For every integer n 2, n is a prime number or can be written uniquely (ignoring ordering) as a product of prime numbers. We ignore the order in which we write the factors: 2(3)(3) = 3(2)(3) If a and b are positive integers, then gcd(a,b) can always be written as a linear combination of a and b, that is, gcd(a,b) = ia + jb for some integers i and j. gcd(420,66) = 6 = 3(420) 19(66)

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Section 3.7Number Theory2 Fundamental Theorem of Arithmetic The values 3 and 19 in gcd(420,66) = 3(420) 19(66) are derived from the successive divisions done by the Euclidean algorithm: 420 = 6 * 66 + 24 66 = 2 * 24 + 18 24 = 1 * 18 + 6 18 = 3 * 6 + 0 Rewriting the first three equations from the bottom up: 6 = 24 * 1 + 18 18 = 66 * 2 + 24 24 = 420 * 6 + 66 Now we use these equations in a series of substitutions: 6 = 24 1 * 18 = 24 1 * (66 2 * 24) (substituting for 18) = 3 * 24 66 = 3 * (420 6 * 66) 66 (substituting for 24) = 3 * 420 19 * 66

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Section 3.7Number Theory3 Fundamental Theorem of Arithmetic THEOREM ON gcd(a, b) Given positive integers a and b, gcd(a,b) is the linear combination of a and b that has the smallest positive value. From the theorem on gcd(a,b), it follows that a and b are relatively prime if and only if there exist integers i and j such that: ia + jb = 1 DEFINITION: RELATIVELY PRIME Two integers a and b are relatively prime if gcd(a,b) 1.

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Section 3.7Number Theory4 Fundamental Theorem of Arithmetic THEOREM ON DIVISION BY PRIME NUMBERS Let p be a prime number such that p ab, where a and b are integers. Then, either p a or p b. To find the unique factorization of 825 as a product of primes, we can start by simply dividing 825 by successively larger primes: 825 = 3 * 275 = 3 * 5 * 55 = 3 * 5 * 5 * 11 = 3 * 5 2 * 11 Doing the same on 455: 455 = 5 * 7 * 13 From these factorizations, we can see that gcd(825, 455) = 5.

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Section 3.7Number Theory5 More on Prime Numbers THEOREM ON SIZE OF PRIME FACTORS If n is a composite number, then it has a prime factor less than or equal to (n) 1/2. Given n = 1021, let’s find the prime factors of n or determine that n is prime. The value of (1021) 1/2 is just less than 32. So the primes we need to test are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. None divides 1021, so 1021 is prime.

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Section 3.7Number Theory6 More on Prime Numbers THEOREM ON INFINITY OF PRIMES (EUCLID) There is an infinite number of prime numbers. Assume that there is a finite number of primes. Let the value of s = the sum of all primes + 1. Therefore, s is not prime. Thus, s is composite and by the fundamental theorem of arithmetic, s can be factored as a product of (some of) the prime numbers. Suppose that p j is one of the prime factors of s, that is, s = p j (m) for some integer m. Then: 1 = s – p 1 p 2 … p k = p j (m) p 1 p 2 … p k = p j (m p 1 … p j 1 p j + 1 …p k ) Therefore, p j 1, which is a contradiction.

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Section 3.7Number Theory7 Euler Phi Function DEFINITION: EULER PHI FUNCTION For n an integer, n 2, the Euler (pronounced “oiler”) phi function of n, (n), is the number of positive integers less than or equal to n and relatively prime to n. ( (n) is pronounced “fee” of n.) For example: (2) = 1 (the number 1) (3) = 2 (the numbers 1, 2) (4) = 2 (the numbers 1, 3) (5) = 4 (the numbers 1, 2, 3, 4) (6) = 2 (the numbers 1, 5) (7) = 6 (the numbers 1, 2, 3, 4, 5, 6)

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Chapter 4 Properties of the integers: mathematical induction Yen-Liang Chen Dept of IM NCU.

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