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Direct Proof and Counterexample III
Lecture 15 Section 3.3 Mon, Feb 12, 2007
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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. Divisibility is a “positive” property.
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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?
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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?
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Units and Zero Definition: An integer u is a unit if u | 1.
The only units are 1 and –1. Definition: 0 is zero.
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Example: Direct Proof Theorem: If a | b and b | c, then a | c. Proof:
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