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

2. 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.

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: