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Direct Proof and Counterexample III

Direct Proof and Counterexample III. Lecture 15 Section 3.3 Mon, Feb 12, 2007. Divisibility. Definition: An integer a divides an integer b if a  0 and there exists an integer c such that ac = b . Write a | b to indicate that a divides b .

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Direct Proof and Counterexample III

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  1. Direct Proof and Counterexample III Lecture 15 Section 3.3 Mon, Feb 12, 2007

  2. Divisibility • Definition: An integer adivides an integer b if a 0 and there exists an integer c such that ac = b. • Write a | b to indicate that a divides b. • Divisibility is a “positive” property.

  3. Prime Numbers • Definition: An integer p is prime if p 2 and the only positive divisors of p are 1 and p. • A prime number factors only in a trivial way: p = 1  p. • Prime numbers: 2, 3, 5, 7, 11, … • Is this a positive property?

  4. Composite Numbers • Definition: An integer n is composite if there exist integers a and b such that a > 1 and b > 1 and n = ab. • A composite number factors in a non-trivial way. • Composite numbers: 4, 6, 8, 9, 10, 12, … • Is this a positive property?

  5. Units and Zero • Definition: An integer u is a unit if u | 1. • The only units are 1 and –1. • Definition: 0 is zero.

  6. Example: Direct Proof • Theorem: If a | b and b | c, then a | c. • Proof:

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