1. 1 Properties of Integers Objectives At the end of this unit, students should be able to: State the division algorithm Apply the division algorithm Find the gcd using Euclidean algorithm Use the gcd to represent a linear combination of two integers Find the least common multiple Represent an integer as a product of primes

2. 2 Contents Division algorithm Greatest common divisor Fundamental Theorem of Arithmetic

3. 3 Division Algorithm Definition 4.1 Given a, b  Z and b  0, b divides a, written as b | a, if there is an integer n such that a = bn. b is a divisor of a, or a is a multiple of b. E.g 2 | 8, 3 | 15, etc. 3 I 7

4. 4 Theorem 4.3 For all a, b, c  Z (set of integers), a. 1|a and a|0. b. [(a | b)  (b | a)]  a =  b. c. [(a | b)  (b | c)]  a | c. d. a | b  a | bx for all x  Z.

5. 5 Theorem 4.3 (cont.) e. If x = y + z, for some x, y, z  Z, and a divides two of the three integers x, y, and z, then a divides the remaining integer. f. [(a | b)  (a | c)]  a | (bx + cy), for all x, y  Z. (The expression bx + cy is called a linear combination of b, c.) g. For 1  i  n, let c i  Z. If a divides each c i, then a| (c 1 x 1 + c 2 x 2 + … + c n x n ), where x i  Z for all 1  i  n.

6. 6 Proof (part f) If a|b and a|c, then b = am and c = an, for some m, n  Z. So bx + cy = (am)x + (an)y = a(mx + ny) Since bx + cy = a(mx + ny), with mx + ny  Z, it follows that a | (bx + cy).

7. 7

8. 8 Question ???

9. 9 Prime and Composite Numbers

10. 10 Prime Numbers LEMMA 4.1 If n  Z + and n is composite, then there is a prime p such that p | n. E.g 5 | 20 (5 is a prime and 20 a composite) Theorem 4.4 There are infinitely many primes.

11. 11 Division Algorithm Theorem 4.5 The Division Algorithm. If a, b  Z, with b > 0, then there exist unique q, r  Z with a = qb + r, 0  r < b. q: quotient r: remainder b: divisor a: dividend.

12. 12

13. 13

14. 14

15. 15 Greatest Common Divisor Objectives: On completion this unit, you should be able to: 1. Find the gcd using Euclidean algorithm 2. Use the gcd to represent a linear combination of two integers 3. Find the least common multiple 4. Represent an integer as a product of primes

16. 16 Contents Greatest common divisor Least common multiple Fundamental Theorem of Arithmetic

17. 17 Common Divisor c is a common divisor of a and b then c | b and c | b e.g. 3 | 6 and 3 | 9 3 is a common divisor of 6 and 9. 1 | 42 and 1 | 70 2 | 42 and 2 | 70 7 | 42 and 7 | 70 14 | 42 and 14 | 70 (1,2, 7, 14) are common divisors. The greatest common divisor (gcd) is 14.

18. 18 Greatest Common Divisor Definition 4.3 Let a, b  Z, where either a  0 or b  0. Then c  Z + is called a greatest common divisor of a, b if i) c|a and c|b (that is, c is a common divisor of a, b), and ii) for any common divisor d of a and b, we have d|c. (i.e. d ≤ c)

19. 19 Greatest Common Divisor Theorem 4.6 For all a, b  Z +, there exists a unique c  Z + that is the greatest common divisor of a, b. gcd(a, b) = gcd(b, a) For each a  Z, if a  0, then gcd(a, 0) = |a|. when a, b  Z +, we have gcd(–a, b) = gcd(a, –b) = gcd(– a, – b) = gcd(a, b) gcd(0, 0)  not defined and is of no interest to us.

20. 20 Relatively Prime integers From Theorem 4.6, not only does gcd(a, b) exist but that gcd(a, b) is also the smallest positive integer we can write as a linear combination of a and b. Integers a and b are called relatively prime when gcd(a, b) = 1 that is, when there exist x, y  Z with ax + by = 1.

21. 21

22. 22

23. 23

24. 24 Question ???

25. 25

26. 26

27. 27 Activity Find gcd(72,63), gcd(330,156)

28. 28 Least Common Multiple Definition 4.4 For a, b, c  Z +, c is called a common multiple of a, b if c is a multiple of both a and b. Furthermore, c is the least common multiple of a, b if it is the smallest of all positive integers. We denote c by lcm(a, b).

29. 29

30. 30

31. 31

32. 32 Fundamental Theorem of Arithmetic

33. 33

34. 34

35. 35

36. 36

37. 37