Lecture 5

1. Lecture 5 Learning Objectives To apply division algorithm To apply the Euclidean algorithm

2. Algorithms • An algorithm is a systematic procedures (instructions) for calculation. • Algorithms are basic to computer programs. Essentially, a program implements one or more algorithms. Therefore, algorithmic complexity is important. • In this Lecture, we will study a few algorithms: • Division algorithm • Euclidean algorithm • Primality testing

3. RMIT University; Taylor's College Activity 1 Pick a integer between 0 to 15 Is it on Card A? Is it on Card B? Is it on Card C? Is it on Card D?

4. RMIT University; Taylor's College Activity 2 • Write a set of instructions (algorithms) to write all the integers from 0 to 10.

5. RMIT University; Taylor's College Algorithm example 1 • Step 1: Set • Step 2: • Step 3: • Step 4:, stop • Step 5: Go to Step 2

6. The Division Algorithm

7. The Division Algorithm • For any integer , wecan represent a in the form of where . • a – integer • b – integer> 0 • q – quotient • r – remainder

8. RMIT University; Taylor's College Algorithms • The process of expressing a in this way is the application of the division algorithm • Essentially this says that we can divide one integer by another if the latter is positive, and that we get a quotient and a remainder

9. RMIT University; Taylor's College The Division Algorithm • Write the following integers in the form of

10. The Division Algorithm • If a > 0, then (floor of a/b) • Example: a = 31, b = 7 • So a = bq + r gives 31 = 7 ∙ 4 + 3 • Given a, b: Valid input requires a, b to be integers and b > 0

11. The Euclidean Algorithm

12. Factors (or Divisors) and Multiple Let a, b and c be integers. • Suppose that ab = c. . • We say that c is a multiple of aand of b. • Also, a and b are divisors or factors of c. • Example: • 15 is the multipleof 3 and of 5. • 3 and 5 are divisors(factors) of 15.

13. Common Factor Let m, n be positive integers. • A positive integer qis a common factor or common divisor of m and n if it divides (is a divisor, or factor, of) both of them • Examples: • What is the common factor for 16 and 24 • What is the common factor for 15 and 30

14. RMIT University; Taylor's College Common Multiple • A positive integer pis a common multiple of m and n if it is a multiple of both of them • Examples: • Which of the following is the common multiple of 3 and 6? • 15 • 18 • 24 • 27 • Which of the following is the common multiple of 4 and 9? • 36 • 54 • 72 • 108

15. Greatest Common Divisor (GCD) Let m, n be positive integers. • The GCD (greatest common divisor) of m and n is the greatest number which is a common divisor of both of them • It’s also called the highest common factor or HCF

16. Example 1 What is the GCD of 18 and 24? ? gcd (18, 24) = 6 There is a systematic procedure for getting the GCD. It’s the Euclidean algorithm.

17. Least Common Multiple • Given integers m and n, their least common multiple (LCM)is the smallest number which is a multiple of them both • Examples: • What is the LCM of 8 and 6? • What is the LCM of 3 and 4? The least common multiple of 2 positive integers equals their product divided by their greatest common divisor

18. Euclidean Algorithm • We can get the gcd by using the Euclidean algorithm. • This involves repeated application of the division algorithm: a = bq + r • Euclidean Algorithm When the remainder becomes zero, we look back to the previous remainder, rn+1. This must be the gcd of a and b.

19. RMIT University; Taylor's College Example 2 gcd (96, 22) = ? The last nonzero remainder was 2. Therefore, gcd (96, 22) = 2. 96 = 4 ∙ 22 + 8 22 = 2 ∙ 8 + 6 8 = 1 ∙ 6 + 2 6 = 3 ∙ 2 No remainder

20. RMIT University; Taylor's College Example 3 gcd (63, 256) = ? The last nonzero remainder was 1. Therefore, gcd (63, 256) = 1. 256 = 4 ∙ 63 + 4 63 = 15 ∙ 4 + 3 4 = 1 ∙ 3 + 1 3 = 3 ∙ 1 No remainder

21. Extension to the Euclidean Algorithm • If d = gcd(m, n) then d can be expressed as a linear combination d = xm + yn of m and n, where x and y are integers • To find x and y, we work back through the steps of the Euclidean algorithm from bottom to top

22. Example 4 • It can be shown that gcd(22, 96) = 2: 96 = 4 ∙ 22 + 8 22 = 2 ∙ 8 + 6 8 = 1 ∙ 6 + 2 6 = 3 ∙ 2 • Now we want to express 2 as a linear combination 2 = x(22) + y(96). We use the second-last line to make 2 the subject of the equation: 2 = 8 – 1 ∙ 6 • Next we use the third-last line to express 6 in terms of 22 and 8, substituting this into the equation we’ve just produced: 2 = 8 – 1 ∙ 6 = 8 – 1 ∙ (22 – 2 ∙ 8) = 8 – 1 ∙ 22 + 1 ∙ 2 ∙ 8 = 3 ∙ 8 – 1 ∙ 22

23. Example 4 (cont.) • Finally we use the fourth-last line to express 8 in terms of 96 and 22, substitution this into our most recent equation 2 = 3 ∙ 8 – 1 ∙ 22 2= 3 ∙ (96 – 4 ∙ 22) – 1 ∙ 22 2= 3 ∙ 96 – 3 ∙ 4 ∙ 22 – 1 ∙ 22 2= 3 ∙ 96 – 13 ∙ 22 x=3, y=-4

24. Example 5 It can be shown that the gcd of 63 and 256 equals 1: 256 = 4 ∙ 63 + 4 63 = 15 ∙ 4 + 3 4 = 1 ∙ 3 + 1 3 = 3 ∙ 1 Then we work upwards from the second-last line, as follows: 1 = 4 - 1 ∙ 3 = 4 – 1 ∙ (63 – 15 ∙ 4) = 4 - 1 ∙ 63 + 1 ∙ 15 ∙ 4 = 16 ∙ 4 – 1 ∙ 63 = 16 ∙ (256 – 4 ∙ 63) – 1 ∙ 63 = 16 ∙ 256 – 64 ∙ 63 - 1 ∙ 63 = 16 ∙ 256 – 65 ∙ 63 • So 1 = 16 ∙ 256 – 65 ∙ 63. • In this example, 63 and 256 are relatively prime.

25. RMIT University; Taylor's College Prime Numbers • A prime number is an integer ≥ 2 which has no factors except itself and 1 • Prime numbers: 2, 3, 5, 7, … • Prime numbers play a vital role in coding and cryptography • We say two positive integers are relatively prime (in relation to each other) if their gcd equals 1 • So 63 and 256 are relatively prime (to each other), even though neither of them is a prime number

26. BBC News (online) dated 5th December 2001 http://news.bbc.co.uk/2/hi/science/nature/1693364.stm, accessed 1st September 2009

27. Prime Number • How do you determine a prime number? • PrinciplePrime number is an integer that is only divisible by 1 and the integer itself. • If an integer is divisible by integers other than 1 and itself, it is not a prime number. • Example:Is 357 a prime number?Is 271 a prime number?

28. Prime Number • Is 357 a prime number? • Solution:is 357 a prime number? • 357 is not divisible by 2. • 119357 is divisible by 3, it is not a prime number.

29. Prime Number • Is 271 a prime number? • Solution: • 271 is not divisible by 2. • 271 is not divisible by 3. • 271 is not divisible by 5 • …Continue to divide 271 by all the odd integers. We find that 271 is not divisible by any integer. • Conclusion: 271 is prime number.

30. Prime Number • Algorithm to determine a prime number. • Assignt = an integer to be tested,d= integer as divisor • Step 1: Set d = 2 • Step 2: if t mod d = 0, then t is not a prime number. Stop. • Step 3: Set d = 3 • Step 4: if t mod d = 0, then t is not a prime number. Stop. • Step 5: d = d + 2 • Step 6: if go to step 4 • Step 7: t is a prime number.

31. Activity 3 • Write down the first ten prime numbers. • 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19 • 23 • 29

32. The End

33. RMIT University; Taylor's College

34. Prime Number • Algorithm to determine the first 10 prime numbers. • Assignt = an integer to be tested,d = integer as divisor • Step 1: Set t = 2 • Step 2: Set d = 2 • Step 3: Do step 4 to step if d < t. Else, t is a prime number. Go to Step . • Step 4: if t mod d = 0, then t is not a prime number. Stop. • Step 5: Set d = 3 • Step 4: if t mod d = 0, then t is not a prime number. Stop. • Step 5: d = d + 2 • Step 6: if go to step 4 • Step 7: t is a prime number.