In this lecture

1. In this lecture • Number Theory ● Rational numbers ● Divisibility • Proofs ● Direct proofs (cont.) ● Common mistakes in proofs ● Disproof by counterexample

2. Common mistakes in proofs • Arguing from examples • Using same letter to mean two different things • Jumping to a conclusion (without adequate reasons)

3. Disproof by counterexample • To disprove statement of the form “xD if P(x) then Q(x)”, find a value of x for which ● P(x) is true and ● Q(x) is false. • Ex: For any prime number a, a2-1 is even integer. Counterexample: a=2.

4. Rational Numbers • Definition: r isrationaliff  integers a and b such that r=a/b and b≠0. • Examples: 5/6, -178/123, 36, 0, 0.256256256… • Theorem: Every integer is a rational number.

5. Properties of Rational Numbers • Theorem: The sum of two rational numbers is rational. • Proof: Supposer and s are rational numbers. Thenr=a/b and s=c/d for some integers a,b,c,ds.t.b≠0, d≠0.(by definition) So (by substitution) (by basic algebra) Let p=ad+bcand q=bd. Thenr+s=p/q wherep,q Z and q≠0. Thus,r+s is rational by definition. ■

6. Types of Mathematical Statements • Theorems: Very important statements that have many and varied consequences. • Propositions: Less important and consequential. • Corollaries: The truth can be deduced almost immediately from other statements. • Lemmas: Don’t have much intrinsic interest but help to prove other theorems.

7. Divisibility • Definition: Forn,d Z and d≠0 we say that n is divisible by d iffn=d·k for some k Z . • Alternative ways to say: n is a multiple of d , d is a factor of n , d is a divisor of n , d divides n . • Notation: d | n . • Examples: 6|48, 5|5, -4|8, 7|0, 1|9 .

8. Properties of Divisibility • For xZ, 1|x . • For xZ s.t. x≠0, x|0 . • An integer x>1 is prime iff its only positive divisors are 1 and x . • For a,b,cZ, if a|b and a|c then a|(b+c) . • Transitivity: For a,b,cZ, if a|b and b|c then a|c .

9. Divisibility by a prime • Theorem: Any integer n>1 is divisible by a prime number. • Sketch of proof: Division into cases: ●If n is prime then we are done (since n | n). ●If n is composite then n=r1·s1where r1,s1Z and 1<r1<n,1<s1<n. (by definition of composite number) (Further)division into cases: ♦If r1 is prime then we are done (since r1 |n). ♦If r1 is composite then r1=r2·s2where r2,s2Z and 1<r2<r1,1<s2<r1.

10. Divisibility by a prime • Sketch of proof (cont.): Since r1|n and r2|r1 then r2 |n (by transitivity). Continuing the division into cases, we will get a sequence of integers r1 , r2 , r3 ,…, rk such that 1< rk< rk-1<…< r2< r1<n ; rp |n for each p=1,2,…,k ; rk is prime. Thus, rk is a prime that divides n. ■

11. Unique Factorization Theorem • Theorem: For integer n>1,  positive integer k, distinct prime numbers , positive integers s.t. , andthis factorization is unique. • Example: 72,000 =