Discrete Mathematics Chapter 4 Number Theory

Chapter 4 Number Theory Integers Integers are whole numbers, without any fractional or decimal components. Example: 8, 21, 8765, –34, 0. They are in opposite to real numbers, which posses decimal components. Example: 8.0, 34.25, 0.02, 3.2222

Chapter 4 Number Theory Division of Integers Suppose a and b are integers, a  0, then a divides b without remaining if there exists an integer c such that b = ac. Notation: a | b if b = ac, c  Z and a  0. Example: 4 | 12 because 12/4 = 3 (integer) or 12 = 4  3. 4 | 13 because 13/4 = 3.25 (not integer).

Chapter 4 Number Theory Euclidean Theorem Suppose m and n are integers, n > 0. If m is divided by n then there exists a unique integer q (quotient) and r (remainder), such that m = nq + r where 0  r < n. m is called dividend, while d is called divisor. Example: 1987/97 = 20, remaining 47 1987 = 9720 + 47 25/7 = 3, remaining 4 25 = 73 + 4 –25/7 = –4, remaining 3 –25 = 7(–4) + 3 But not –25 = 7(–3) – 4, because the remainder will be r = –4, while the condition is 0  r < n)

Greatest Common Divisor (GCD) Chapter 4 Number Theory Greatest Common Divisor (GCD) Suppose a and b are non-zero integers. The Greatest Common Divisor (GCD) of a and b is the greatest possible integer d such that d | a and d | b. In this case, it can be written as GCD(a,b) = d. Example: Determine GCD(45,36) ! Divisors of 45: 1, 3, 5, 9, 15, 45. Divisors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Common divisors of 45 and 36 are 1, 3, 9. For the enumeration above, it can be concluded that GCD(45,36) = 9.

Greatest Common Divisor (GCD) Chapter 4 Number Theory Greatest Common Divisor (GCD) Suppose m and n are integer, n > 0, such that m = nq + r, 0  r < n. Then GCD(m,n) = GCD(n,r). Example: Take the value m = 66, n = 18, 66 = 183 + 12 then GCD(66,18) = GCD(18,12) = 6.

Chapter 4 Number Theory Euclidean Algorithm Euclid (around 300 BC) is a Greek mathematician, who wrote this algorithm in his book titled, “Element.” The objective of this algorithm is to find the GCD of two integers.

Chapter 4 Number Theory Euclidean Algorithm If m and n are non-negative integers where m  n, and suppose r0 = m and r1 = n. Perform the following divisions in sequence to obtain: r0 = r1q1 + r2 0  r2  r1, r1 = r2q2 + r3 0  r3  r2, ri–2 = ri–1qi–1 + ri 0  ri  ri–1, ri–1 = riqi + 0 . . . . . . According to the statement on the previous section, GCD(m,n) = GCD(r0,r1) = GCD(r1,r2) = … = GCD(ri–2,ri–1) = GCD(ri–1,ri) = GCD(ri,0) = ri Thus, GCD of m and n is the last non-zero remainder of the above sequence of disions, namely ri.

Chapter 4 Number Theory Euclidean Algorithm Given two non-negative integers m and n (m  n), the following Euclidean Algorithm will find the greatest common divisor of m and n. Step 1: If n = 0 then m is the GCD(m,n); Finish. If n  0, proceed to Step 2. Step 2: Divide m with n and obtain r as the remainder. Step 3: Replace m with n, and n with r, then loop back to Step 1.

Euclidean Algorithm Example: Chapter 4 Number Theory Euclidean Algorithm Example: Take m = 80, n = 12, so the condition that m  n is fulfilled. 80 = 126 + 8 12 = 81 + 4 8 = 42 + 0 n = 0  m = 4 is the last non-zero remainder GCD(80,12) = 4; Finish.

Chapter 4 Number Theory Linear Combination GCD(a,b) can be expressed as a linear combination of a and b with the multiplying coefficients that can be freely chosen. Example: GCD(80,12) = 4, then 4 = (–1)80 + (7)12, where –1 and 7 are coefficients that can be freely chosen. Suppose a and b are positive integers, then there exist integers m and n such that GCD(a,b) = ma + nb.

Chapter 4 Number Theory Linear Combination Example: Express GCD(312,70) = 2 as the linear combination of 312 and 70! Applying Euclidean Algorithm: 312 = 470 + 32 (1) 70 = 232 + 6 (2) 32 = 56 + 2 (3) 6 = 32 + 0 (4) Thus, GDC(312,70) = 2 Rearrange (3) to 2 = 32 – 56 (5) Rearrange (2) to 6 = 70 – 232 (6) Insert (6) to (5) so that 2 = 32 – 5(70 – 232) = 132 – 570 + 1032 = 1132 – 570 (7) Rearrange (1) to 32 = 312 – 470 (8) Insert(8) to (7) so that 2 = 1132 – 570 = 11(312 – 470) – 570 = 11312 – 4970 Thus, GCD(312, 70) = 2

Chapter 4 Number Theory Modulo Arithmetics Suppose a is an arbitrary integer and m is a positive integer, then a mod m yields the remainder if a is divided by m a mod m = r such that a = mq + r, with 0  r < m The result of modulo m lies within the set {0,1,2,…,m–1}

Chapter 4 Number Theory Congruence Examine that 38 mod 5 = 3 and 13 mod 5 = 3, then it can be written that 38  13 (mod 5). Pronounce: 38 is congruent with 13 in modulo 5. Suppose a and b are integers and m > 0. If m divides a – b without remainder, then a  b (mod m). If a is not congruent with b in modulo m, then it is written as a  b (mod m).

Congruence Example: 17  2 (mod 3) Chapter 4 Number Theory Congruence Example: 17  2 (mod 3)  3 divides 17–2 = 15 without remainder –7  15 (mod 11)  11 divides –7–15 = –22 without remainder 12  2 (mod 7)  7 cannot divide 12–2 = 10 –7  15 (mod 3)  3 cannot divide –7–15 = –22

Congruence a  b (mod m) can be written as a = b + km (k integer). Chapter 4 Number Theory Congruence a  b (mod m) can be written as a = b + km (k integer). Example: 17  2 (mod 3)  17 = 2 + 53 –7  15 (mod 11)  –7 = 15 + (–2)11 a mod m = r can also be written as a  r (mod m). Example: 23 mod 5 = 3  23  3 (mod 5) 14 mod 8 = 6  14  6 (mod 8) –41 mod 9 = 4  –41  4 (mod 9) –39 mod 13 = 0  –39  0 (mod 13)

Congruence Theorem Suppose m is a positive integer. Chapter 4 Number Theory Congruence Theorem Suppose m is a positive integer. If a  b (mod m) and c is an arbitrary integer, then (a + c)  (b + c) (mod m) ac  bc (mod m) ap  bp (mod m) , p non-negative If a  b (mod m) and c  d (mod m), then (a + c)  (b + d) (mod m) ac  bd (mod m)

Chapter 4 Number Theory Congruence Example: Suppose 17  2 (mod 3) and 10  4 (mod 3), then according to the Congruence Theorem, 17 + 5  2 + 5 (mod 3)  22  7 (mod 3) 175  25 (mod 3)  85  10 (mod 3) 17 + 10  2 + 4 (mod 3)  27  6 (mod 3) 1710  24 (mod 3)  170  8 (mod 3)

Chapter 4 Number Theory Prime Numbers A positive integer p (p > 1) is called a prime number if its divisors are only 1 and p. For example, 23 is a prime number, because it can only be divided by 1 and 23 to get no remainder. Numbers which are not prime numbers are called composite numbers. For example, 20 is a composite number, because 20 is divisible by 2, 4, 5, and 10, besides by 1 and 20 itself.

Chapter 4 Number Theory Relatively Prime Two integers a and b are said to be relatively prime if they do not have any common factors other than 1, or, GCD(a,b) = 1. Example: 20 and 3 are relatively prime, since GCD(20,3) = 1. 7 and 11 are relatively prime, since GCD(7,11) = 1. 20 and 5 are not relatively prime, since GCD(20,5) = 5 ≠ 1.

Chapter 4 Number Theory Relatively Prime If a and b are relatively prime, then there exist integers m and n such that ma + nb = 1. Example: 20 and 3 are relatively prime because GCD(20,3) =1, so that it can be written that 220 + (–13)3 = 1 (m = 2, n = –13). 20 and 5 are not relatively prime because GCD(20,5) ≠ 1, and thus 20 and 5 cannot be written in the form of m20 + n5 = 1.

Linear Congruence The linear congruence is in the form of: Chapter 4 Number Theory Linear Congruence The linear congruence is in the form of: ax  b (mod m), where m > 0, a and b are arbitrary integers, and x is any integer. The solution can be found in the way: ax = b + km  x = (b + km) / a Try each value of k = 0, 1, 2, … and k = –1, –2, … that delivers integer value for x.

Chapter 4 Number Theory Linear Congruence Example: Determine the solutions for 4x  3 (mod 9) ! 4x  3 (mod 9)  x = (3 + k9 ) / 4 k = 0  x = (3 + 09) / 4 = 3/4  not a solution k = 1  x = (3 + 19) / 4 = 3  a solution k = 2  x = (3 + 29) / 4 = 21/4  not a solution k = 3, k = 4  no solution k = 5  x = (3 + 59) / 4 = 12  a solution … k = –1  x = (3 – 19) / 4 = –6/4  not a solution k = –2  x = (3 – 29) / 4 = –15/4  not a solution k = –3  x = (3 – 39) / 4 = –6  a solution k = –7  x = (3 – 79) / 4 = –15  a solution The set of solutions is: {3, 12, …, –6, –15, …}.

Chapter 4 Number Theory Linear Congruence Example: Determine the solutions for 2x  3 (mod 4) ! 2x  3 (mod 4)  x = (3 + k4 ) / 2 Because k4 is always an even number, then 3 + k4 will always be an odd number. If an odd number is divided by 2, then the result will be a decimal number (never be an integer). Thus, there is no value of x that can be the solution of 2x  3 (mod 4).

Chapter 4 Number Theory Homework 6 Determine GCD(216,88) and express the GCD as a linear combination of 216 and 88.

Chapter 4 Number Theory Homework 6A Determine GCD(74,16) and express the GCD as a linear combination of 74 and 16. Determine the solutions for 5x  7 (mod 11) !