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Chapter 3

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

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Chapter 3

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

  2. 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.

  3. 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=2252 641=641 999=3*3*3*37=33*37 1024=2*2*2*2*2*2*2*2*2*2=210

  4. 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.

  5. 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, p1, p2, … , pn. Let Q=

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

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

  8. Greatest Common Divisors • Definition 4: The integers a1,a2 …,an are pairwise relatively primeif gcd(ai , aj)=1 whenever 1≦i<j ≦n. • Example 13: determine whether the integers 10 , 17 and 21 are pairwise relatively prime and whether the integers 10 , 19 and 24 are pairwise relatively prime. • Example 14: Because the prime factorizations of 120 and 500 are 120=23*3*5 and 500=22*53,the greatest common divisor is gcd(120,500)=2 min(3 , 2) 3 min(1 , 0) 5 min(1,3)=223051=20

  9. Least Common Multiples • Definition 5: Theleast 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 233572 and 2433?

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

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