# Unit 8 -Elementary Number Theory.pptx - MATH221 Mathematics...

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MATH221 Mathematics for Computer Science Unit 8 Elementary Number Theory 1
2 OBJECTIVES Understand the definition of divisibility. Understand apply the Quotient-Remainder Theorem. Understand and be able to find the Greatest Common Divisor. Understand and apply the Euclidean Algorithm. Understand the concept of the Fundamental Theorem of Arithmetic.
3 Divisibility If n and d are integers and d 0, then n is divisible by d if and only if n = d k for some k . We write d | n and say that d divides n (n is divisible by d) . In logic notation, the definition of divisibility is written d | n k , n = dk .
4 Divisibility Alternatively, if d | n k , n = dk . We say that n is a multiple of d , or d is a factor of n , or d is a divisor of n , or d divides n .
5 Discussion: 1. Is -16 a divisor of 32? Yes. 32 = -16 -2. 2. If l and l 0, does l | 0? Yes. 0 = l k for some k .
6 Discussion: 3. Find all values of a such that a | 1. a | 1 k , 1 = ak . For a = 1 and k = 1, ak = 1. For a = -1 and k = -1, ak = 1. Therefore, a = -1 or 1.
7 Discussion: 4. What is the relationship between a and b if a | b , and b | a , a , b ? Now, a | b k , b = ak (1) b | a l , a = bl (2) Substitute (2) into (1), b = ( bl ) k Divide both sides by b , 1 = lk Since l , k , l = k = 1. or l = k = -1. Hence, a = b .
8 Discussion: 5. If a , b , is 3 a + 3 b divisible by 3? Now, 3 a + 3 b = 3( a + b ) = 3 s where s = a + b and s . Hence, 3 | 3 a + 3 b , that is, 3 a + 3 b is divisible by 3.
9 Discussion: 6. If a , b , c , x , y . If b | a and b | c , does b | ( ax + cy )? Why? Now, b | a k , a = bk (1) b | c l , c = bl (2) (1) x , ax = bkx (3) (2) y , cy = bly (4) (3) + (4), ax + cy = bkx + bly = b ( kx + ly ) = bm where m = kx + ly , m . Hence, b | ( ax + cy ).
10 Discussion: 7. If a , b , is it true that a | b implies a b ? Now, a | b k , b = ak . Since a , b 1, so k 1. Multiply both sides of the inequality by a , ka 1 a . Hence, b a . That is, a b Therefore, if a , b , it is true that a | b implies a b .
11 Theorem - Transitivity of Divisibility For all integers a , b and c , if a | b and b | c , then a | c . Note: After going through the details of the proof for this theorem, for the remaining theorems, lemma, etc., we shall focus on the application. For their proofs, please refer to the supplementary notes.
12 Transitivity of Divisibility: Proof We know that a | b k , b = ak (1) b | c l , c = bl (2) Show that a | c , that is, find m such that c = ma . Now, c = bl by (2) = ( ak ) l by (1) = ( kl ) a by associativity and commutativity Let m = kl , then m (since is a closed operation on ).