LECTURE 5 Learning Objectives  To apply division algorithm  To apply the Euclidean algorithm

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

Activity 1 Card RMIT University; Taylor's College Card Card Card 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?

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

Algorithm example 1 RMIT University; Taylor's College

The Division Algorithm

RMIT University; Taylor's College Algorithms

RMIT University; Taylor's College The Division Algorithm

If a > 0, then (floor of a/b) Example: a = 31, b = 7 So a = bq + r gives 31 = 7 ∙ Given a, b: Valid input requires a, b to be integers and b > 0

The Euclidean Algorithm

Factors (or Divisors) and Multiple

Common Factor Let m, n be positive integers. A positive integer q is a common factor or common divisor of m and n if it divides (is a divisor, or factor, of) both of them Examples: 1. What is the common factor for 16 and What is the common factor for 15 and 30

Common Multiple A positive integer p is a common multiple of m and n if it is a multiple of both of them Examples: 1. Which of the following is the common multiple of 3 and 6? Which of the following is the common multiple of 4 and 9? RMIT University; Taylor's College

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

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.

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: 1. What is the LCM of 8 and 6? 2. 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

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, r n+1. This must be the gcd of a and b.

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

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

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

Example 4 It can be shown that gcd(22, 96) = 2: 96 = 4 ∙ = 2 ∙ = 1 ∙ = 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 ∙ ∙ 2 ∙ 8 = 3 ∙ 8 – 1 ∙ 22

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

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

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 RMIT University; Taylor's College

accessed 1 st September 2009 BBC News (online) dated 5 th December 2001

Prime Number

Activity 3 Write down the first ten prime numbers

The End

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Prime Number