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Chapter 3 3.5 Primes and Greatest Common Divisors ‒Primes ‒Greatest common divisors and least common multiples 1

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Primes Definition 1: A positive integer p greater than 1 is called prime if the only positive factors of p are 1 and p. A positive integer that is greater than 1 and is not prime is called composite. Remark: The integer n is composite if and only if there exists an integer a such that a|n and 1< a < n. Example 1: The integer 7 is prime because its only positive factors are 1 and 7, whereas the integer 9 is composite because it is divisible by 3. 2

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Primes Theorem 1: The fundamental theorem of arithmetic Every positive integer greater than 1 can be written uniquely as a prime or as the product of two or more primes where the prime factors are written in order of nondecreasing size. Example 2: The prime factorizations of 100, 641, 999 and 1024 are given by 100=2*2*5*5=2 2 5 2 641=641 999=3*3*3*37=3 3 *37 1024=2*2*2*2*2*2*2*2*2*2=2 10 3

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Primes Theorem 2: If n is a composite integer, then n has a prime divisor less than or equal to. Example 3: Show that 101 is prime. Example 4: Find the prime factorization of 7007. 4

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Primes Theorem 3: There are infinitely many primes. Proof: We will prove this theorem using a proof by contradiction. We assume that there are only finitely many primes, p 1, p 2, …, p n. Let Q= 5

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Greatest Common Divisors Definition 2: Let a and b be integers, not both zero. The largest integer d such that d|a d|b is called the greatest common divisor of a and b. The greatest common divisor of a and b is denoted by gcd(a,b). Example 10: what is the greatest common divisor of 24 and 36? 6

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Greatest Common Divisors Definition 3 : The integers a and b are relatively prime if their greatest common divisor is 1. Example 12: Prove that y the integers 17 and 22 are relatively prime. 7

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Greatest Common Divisors Definition 4: The integers a 1,a 2 …,a n are pairwise relatively prime if gcd(a i, a j )=1 whenever 1 ≦ i

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Least Common Multiples Definition 5: The least common multiple of the positive integers a and b is the smallest positive integer that is divisible by both a and b. The least common multiple of a and b is denoted by lcm(a, b). Example 15: What is the gcd and lcm of 2 3 3 5 7 2 and 2 4 3 3 ? 9

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Greatest Common Divisors and Least Common Multiples Theorem 5: Let a and b be positive integers. Then ab = gcd(a,b)* lcm(a, b) 10

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