COMP239: Mathematics for Computer Science II. Prof. Chadi Assi EV7.635


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1 COMP239: Mathematics for Computer Science II Prof. Chadi Assi EV7.635
2 The Euclidean Algorithm
3 The Euclidean Algorithm Finding the GCD of two numbers using prime factorization is inefficient and time consuming Illustration of the Euclidean Algorithm: Determine GCD (91, 287) : 287 = Observe that any divisor of 287 and 91 must also be a divisor of 14 Observe, too, that any divisor of 91 and 14 is also a divisor of 287
4 The Euclidean Algorithm The GCD (287, 91) is the same as the GCD (91, 14) Therefore, one needs to find only GCD (91, 14). Similarly: 91 = and GCD (91, 14) = GCD (14, 7) 14 = 7 2 and GCD (14, 7) = 7 Through successive divisions, one can find the GCD of two integers (no need for prime factorization!)
5 The Euclidean Algorithm Lemma 1: Let a = bq + r, all are integers, then: GCD(a, b) = GCD(b, r) Proof (in class): If we show that the common divisors of a and b are the same as the common divisors of b and r, then GCD (a, b) = GCD (b, r)
6 The Euclidean Algorithm Suppose a and b are positive integers, a b Let r 0 =aand r 1 = b Perform successive divisions: r 0 =r 1 q 1 +r 2 0 r 2 < r 1 r 1 =r 2 q 2 +r 3 0 r 3 < r 2.. r n2 =r n1 q n1 +r n 0 r n < r n1 r n1 = r n q n We know from Lemma 1 that: GCD (a, b) = GCD (r 0, r 1 ) = GCD (r 1, r 2 ) =. = GCD (r n2, r n1 ) = GCD (r n1, r n ) = GCD (r n, 0) = r n
7 The Euclidean Algorithm The GCD is the last non zero remainder in the sequence of divisions. Example: Use Euclidean Algorithm to find GCD (111, 201) 201 = = = = = 3 2
8 The Euclidean Algorithm procedure gcd (a, b: positive integers) x := a y := b while y 0 begin r := x mod y x := y y := r end {GCD (a, b) is x}
9 The Euclidean Algorithm Efficiency: Theorem The number of steps of the Euclidean algorithm applied to two positive integers a and b is at most log 2 a + log 2 b.
10 Applications of Number Theory
11 Some Useful Results Theorem a and b are integers, then there exist integers s and t such that: Example: GCD(6, 14) = 2 and 2 = (2) GCD(a, b) = sa + tb
12 Some Useful Results To express the GCD (a,b) as a linear combinations of a and b, work backward through the divisions of the Euclidean algorithm Example: Express GCD(662, 414) = 2 as a linear combination of 662 and 414.
13 Some Useful Results Example: Express GCD(662, 414) = 2 as a linear combination of 662 and 414. The Euclidean Algorithm uses these divisions 662 = = = = = 41 2
14 Some Useful Results Clearly, we can write: 2 = and we also can write 82 = = =
15 Some Useful Results Backsubstitution gives: GCD(662, 414) = 2 = = ( ) 2 = = ( ) = = ( ) 5 = = 662 (5) (8)
16 Some Useful Results Lemma 1: If a, b, c are positive integers and GCD(a, b) = 1 and a bc, then a c. Proof (in class) Lemma 2: (Generalization of Lemma 1) If p is prime and p a 1 a 2 a n, where a i is an integer, then p a i for some i. Proof (in class)
17 Some Useful Results The generalized Lemma can be used in the proof of uniqueness of prime factorizations. Prove that the decomposition of a composite into primes is unique (this is part of the Fundamental Theorem of Arithmetic). Proof (in class)
18 Some Useful Results Recall (Theorem 5, section 3.4) if m is a positive integer and a, b, c are integers. if a b (mod m) then ac bc (mod m) However, dividing both sides of the congruence by the same integer does not necessarily yield to a valid congruence. Example: 14 8 (mod 6) but 14/2 8/2 (mod 6)
19 Some Useful Results Theorem 2: Let m be a positive integer, and a, b, c be integers. If ac bc (mod m) and GCD(c, m) = 1 then a b (mod m). Proof (in class) Example: a = 9, b = 1, c = 3, m = 4
20 Linear Congruence ax b (mod m) is called a linear congruence, m is a positive integer, a, b are integers, x is an integer variable. A solution for such a congruence is to find all integers x that satisfy the congruence. Inverse of a (mod m): If a exists such that a a 1(mod m) we say a is an inverse of a (mod m).
21 Linear Congruence Theorem 3 If a, m are relatively prime integers, m > 1, then an inverse of a modulo m exists and is unique modulo m. Proof (in class)
22 Linear Congruence To find the inverse of a modulo m (assuming a and m are relatively prime), find a linear combination of a and m that is equal to 1; then the coefficient of a is the inverse of a modulo m. To find the linear combination, use the Euclidean algorithm and work backward through the steps.
23 Linear Congruence Example: Find the inverse of 5 modulo 9. GCD(5, 9) = 1 therefore inverse of 5 modulo 9 exists. Use Euclidean algorithm: 9 = = = and 4 = = 5 (9 5 1) 1 = = 5 (2) + 9 (1) (mod 9) Therefore 2 is the inverse of 5 modulo 9. (NOTE: every integer congruent to 2 mod 9 is also an inverse) e.g., 11 is also an inverse of 5 modulo 9 since 11 2 (mod 9)
24 Linear Congruence When the inverse a (of a modulo m) exists, we can easily solve the congruence: ax b (mod m) by multiplying both sides by the inverse a Examples: Solve the linear congruence 5x 3(mod 9) Solve the linear congruence 3x 4(mod 7).
25 System of Linear Congruences Puzzle: There are certain things whose number is unknown. When divided by 3, the remainder is 2, when divided by 5, the remainder is 3 and when divided by 7 the remainder is 2. What will be the number of things?
26 System of Linear Congruences To solve the puzzle, we can write the following system x 2(mod 3) x 3(mod 5) x 2(mod 7) And solve it. Next, we will show that when the moduli of a system of linear congruences are pairwise relatively prime, there is a unique solution of the system modulo the product of the Moduli
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