1. Direct Proof and Counterexample III
Lecture 14
Section 3.3
Thu, Feb 3, 2005

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?
Is there a positive characterization of primes?

4. Composite Numbers
Definition: An integer n is composite if there exist integers a and b such that 1 < a < n and 1 < b < n and n = ab.
A composite number factors in a non-trivial way : n = a  b, a > 1, b > 1.
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:
Let a, b, c be integers and assume a | b and b | c.
Since a | b, there exists an integer r such that ar = b.
Since b | c, there exists an integer s such that bs = c.
Therefore, a(rs) = (ar)s = bs = c.
So a | c.

7. Example: Direct Proof
Theorem: Let a and b be integers. If a | b and b | a, then a = b.
Proof:
Let a and b be integers.
Suppose a | b and b | a.
There exist integers c and d such that ac = b and bd = a.
Therefore, acd = bd = a.

8. Example: Direct Proof Therefore, cd = 1.
Thus, c = d = 1 or c = d = -1.
Then a = b or a = -b.

9. Example: Direct Proof
Corollary: If a, b  N and a | b and b | a, then a = b.
This is analogous to the set-theoretic statement that if A  B and B  A, then A = B.
Preview: This property is called antisymmetry.
If a ~ b and b ~ a, then a = b.

10. Example: Direct Proof
Theorem: Let a, b, c be integers. If a | b and b | a + c, then a | c.
Proof:
Let a, b, and c be integers.
Suppose a | b and b | a + c.
There exist integers r and s such that ar = b and bs = a + c.

11. Example: Direct Proof
Substitute ar for b in the 2nd equation to get (ar)s = a + c.
Rearrange the terms and factor to get a(rs – 1) = c.
Therefore, a | c.

12. Example: Direct Proof Theorem: If n is odd, then 8 | (n2 – 1). Proof:
Let n be an odd integer.
Then n = 2k + 1 for some integer k.
So n2 – 1 = (2k + 1)2 – 1 = 4k2 + 4k
= 4k(k + 1).

13. Example: Direct Proof
Either k or k + 1 is even.
Therefore, k(k + 1) is a multiple of 2.
Therefore, n2 – 1 is a multiple of 8.
Can you think of an alternate, simpler proof, based on the factorization of n2 – 1?

14. Example: Direct Proof Theorem: If n is odd, then 24 | (n3 – n). Proof:?

15. Example: If-and-Only-If Proof
Theorem: Let a, b be integers. Then a | b if and only if a2 | b2.
Proof: ()
Suppose that a | b.
Then ac = b for some c  Z.
So, a2c2 = b2 and therefore, a2 | b2.

16. Example: If-and-Only-If Proof
Suppose that a2 | b2.
Then a2d = b2 for some d  Z.
Now what?

17. The Fundamental Theorem of Arithmetic
Theorem: Let n be a positive integer. Then
n = p1a1p2a2…pkak,
where each pi is prime and each ai is nonnegative. Furthermore, the primes and their exponents are unique.

18. The Fundamental Theorem of Arithmetic
Use the Fundamental Theorem of Arithmetic to complete the previous proof.