Discrete Mathematics 4. NUMBER THEORY Lecture 7 Dr.-Ing. Erwin Sitompul

Discrete Mathematics

4. NUMBER THEORYLecture 7

Dr.-Ing. Erwin Sitompulhttp://zitompul.wordpress.com

7/2Erwin Sitompul Discrete Mathematics

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.

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:(a) 4 | 12 because 12/4 = 3 (integer) or 12 = 4 3.(b) 4 | 13 because 13/4 = 3.25 (not integer).

Division of Integers


Euclidean Theorem 1: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 + rwhere 0 r < n.

Example:(a) 1987/97 = 20, remaining 47

1987 = 9720 + 47(b) 25/7 = 3, remaining 4

25 = 73 + 4(c) –25/7 = –4, remaining 3

–25 = 7(–4) + 3But not –25 = 7(–3) – 4,because the remainder will be r = –4, while the condition is 0 r < n)

Euclidean Theorem


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)


Euclidean Theorem 2: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 = 183 + 12then GCD(66,18) = GCD(18,12) = 6

Greatest Common Divisor (GCD)


The objectiveThis algorithm can be used to find the GCD of two integers.

InventorEuclid (around 300 BC), a Greek mathematician, wrote the algorithm in his book titled, “Element.”

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 Euclidean Theorem 2,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.

…

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.

Euclidean Algorithm1. If n = 0 then m is the GCD(m,n); STOP. If n 0, proceed to Step 2.2. Divide m with n and obtain r as the remainder.3. Replace m with n, and n with r, then loop back to Step 1.

Euclidean Algorithm


Example:Take m = 80, n = 12, so the condition that m n is fulfilled.

80 = 126 + 8

12 = 81 + 4

8 = 42 + 0

n = 0 m = 4 is the last non-zero remainderGCD(80,12) = 4; STOP.

Euclidean Algorithm


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 + 712

Coefficients, can be freely chosen

Linear Combination Theorem:Suppose a and b are positive integers, then there exist integers m and n such that GCD(a,b) = ma + nb.

Linear Combination


Example:Express GCD(312,70) = 2 as the linear combination of 312 and 70!

Solution: Apply Euclidean Algorithm to obtain GCD(312,70) = 2 as follows:

312 = 470 + 32 (1) 70 = 232 + 6 (2) 32 = 56 + 2 (3) 6 = 32 + 0 (4)

Rearrange (3) to 2 = 32 – 56 (5)

Rearrange (2) to 6 = 70 – 232 (6)

Insert (6) to (5) so that2 = 32 – 5(70 – 232) = 132 – 570 + 1032 = 1132 – 570 (7)

Rearrange (1) to32 = 312 – 470 (8)

Insert(8) to (7) so that2 = 1132 – 570 = 11(312 – 470) –

570 = 11312 – 4970

Thus, GCD(312, 70) = 2 = 11312 – 4970

Linear Combination


Modulo ArithmeticsSuppose a is an arbitrary integer and m is a positive integer, then

a mod m yields the remainder if a is divided by ma mod m = r such that a = mq + r,

with0 r < m The result of modulo m lies within the set {0,1,2,…,m–1}


Congruence Take a long on this

38 mod 5 = 3 and 13 mod 5 = 3. then it can be written that 38 13 (mod 5).Pronounced: 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).


CongruenceExample: 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

Example: 17 2 (mod 3) 17 = 2 + 53 –7 15 (mod 11) –7 = 15 + (–2)11

Example: 23 mod 5 = 3 23 3 (mod 5) 6 mod 8 = 6 6 6 (mod 8) 0 mod 12 = 0 0 0 (mod 12) –41 mod 9 = 4 –41 4 (mod 9) –39 mod 13 = 0 –39 0 (mod 13)

a b (mod m) can be written asa = b + km (k integer).

a mod m = r can also be written as a r (mod m).


Congruence Theorem:Suppose m is a positive integer.1. If a b (mod m) and c is an arbitrary integer, then

i. (a + c) (b + c) (mod m)ii. ac bc (mod m)iii. ap bp (mod m) , p non-negative

2. If a b (mod m) and c d (mod m), theni. (a + c) (b + d) (mod m)ii. ac bd (mod m)

Congruence


CongruenceExample: 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) 175 25 (mod 3) 85 10 (mod 3) 17 + 10 2 + 4 (mod 3) 27 6 (mod 3) 1710 24 (mod 3) 170 8 (mod 3)


Prime NumbersA positive integer p (p > 1) is called a prime number if its

divisors are only 1 and p.

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.

Example: 20 is a composite number, because 20 is divisible by 2, 4, 5, and 10, besides by 1 and 20 itself.


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.

Relative Primes

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, 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.

Example: 20 and 3 are relatively prime because GCD(20,3) =1, so that

it can be written that 220 + (–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 ofm20 + n5 = 1.


Inverse of Modulo In real number arithmetics, the inverse of multiplication is

division.

As example, the inverse of 4 is 1/4, because 4 1/4 = 1.

In modulo arithmetics, finding the inverse is somehow more difficult.

If a and m are relatively prime and m > 1, then there exists the inverse of “a modulo m”.

The inverse of “a modulo m” is an integer x such that ax 1 (mod m).


Inverse of ModuloExample: Determine the inverse of 4 (mod 9) !

Solution: Because GCD(4,9) = 1, then the inverse of 4 (mod 9) exists.From the Euclidean Algorithm,

9 = 24 + 1.Rearrange the above equation to –24 + 19 = 1.From the last equation, it can be obtained that –2 is the inverse of 4 (mod 9).

Check that –24 1 (mod 9)

1 24 (mod 9)

1 2 4 (mod 9)

1 8 (mod 9)


Inverse of ModuloRemark: Every integer which is congruent with –2 (mod 9) is also the inverse of 4.

Example: 7 –2 (mod 9) 9 divides 7 – (–2) = 9

without remainder –11 –2 (mod 9) 9 divides –11 – (–2) = –9

without remainder 16 –2 (mod 9) 9 divides 16 – (–2) = 18

without remainder

1 24 (mod 9)

1 2 (mod 9)4

1 1 17 (mod 9) , 11 (mod 9) , 16 (mod 9)4 4 4


Example: Determine the inverse of 17 (mod 7) !

Solution: Since GCD(17,7) = 1, then the inverse of 17 (mod 7) exists.From the Euclidean Algorithm,

17 = 27 + 3 (1) 7 = 23 + 1 (2) 3 = 31 + 0 (3)

Rearrange (2) to 1 = 7 – 23 (4)Rearrange (1) to

3 = 17 – 27 (5)Insert (5) to (4)

1 = 7 – 2(17 – 27) 1 = –217 + 57

Inverse of Modulo

From the last equation, –2 is the inverse of 17 (mod 7). Checking, –217 1 (mod 7)


Example: Determine the inverse of 18 (mod 10) !

Solution: Since GCD(18,10) = 2 ≠ 1, then the inverse of 17 (mod 7) does not exist.

Inverse of Modulo


Linear CongruenceThe 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) / aTry each value of k = 0, 1, 2, … and k = –1, –2, … that

delivers integer value for x.


Linear CongruenceExample: Determine the solutions for 4x 3 (mod 9) !

Solution: 4x 3 (mod 9) x = (3 + k9 ) / 4k = 0 x = (3 + 09) / 4 = 3/4 not a solutionk = 1 x = (3 + 19) / 4 = 3 a solutionk = 2 x = (3 + 29) / 4 = 21/4 not a solutionk = 3, k = 4 no solutionk = 5 x = (3 + 59) / 4 = 12 a solution…k = –1 x = (3 – 19) / 4 = –6/4 not a solutionk = –2 x = (3 – 29) / 4 = –15/4 not a solutionk = –3 x = (3 – 39) / 4 = –6 a solution…k = –7 x = (3 – 79) / 4 = –15 a solution…The set of solutions is: {3, 12, …, –6, –15, …}.


Example: Determine the solutions for 2x 3 (mod 4) !

Solution: 2x 3 (mod 4) x = (3 + k4 ) / 2 Because k4 is always an even number, then 3 + k4 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).

Linear Congruence


Linear CongruenceExample: Find x such that 3x 4 (mod 7) !

Solution: 3x 4 (mod 7) (3)–13x (3)–14 (mod 7) x (3)–14 (mod 7) x –24 (mod 7) x –8 (mod 7)

x 6 (mod 7)

x = {..., –8, –1, 6, 13, 19, ...}

1 1(3) (mod 7)3

y

3 1 (mod 7)y 2y


Application: ISBN ISBN (International Standard Book Number)

ISBN consists of 10 characters, commonly separated by space or minus sign, i.e., 0–3015–4561–9.

ISBN is classified into several codes: Code to identify the language of the book Code to identify the publisher Code that uniquely assigned for the book (i.e. titles) Checksum character or check digit (can be a number of

or alphabet X)


10

0 (mod 11)ii i

ix

The check digit is so chosen that

Application: ISBN

9

mod 11 ii i

ix

check digit

Example: ISBN 0–3015–4561–8 0 : Code for English-language country group using, 3015 : Publisher code 4561 : Item number, title of the book 8 : Check digit. The check digit is obtained as follows: 10 + 23 + 30 + 41 + 55 + 64 + 75 + 86 + 91 = 151Therefore, the check digit is 151 mod 11 = 8.


Example: ISBN 978-3-8322-4066-0

Since January 2007, ISBN contains 13 digits. The way to count the check digit is different. It uses modulo

10.

The check digit will be obtained as follows: 91 + 73 + 81 + 33 + 81 + 33 +

21 + 23 + 41 + 03 + 61 + 63 = 100Thus, the check digit is 100 + x13 0 (mod 10)

x13 = 0

Application: ISBN

13odd digit even digit

3 0 (mod 10)i ii i

x x x

How to check the validity of a credit card number?


Homework 7

Determine GCD(216,88) and express the GCD as a linear combination of 216 and 88.

No.1:

No.2: Given the ISBN-13: 978-0385510455, check whether the code is valid or not. Hint: Verify the check digit of the ISBN numbers.


Homework 7New

Determine the solutions for 5x 7 (mod 11) !No.1:

No.2: Given the ISBN-10: 0072880082, check whether the code is valid or not. Hint: Verify the check digit of the ISBN numbers.

No.3: Voluntary for additional 20 points

The ISBN-13: 978-007289A054 is valid. What will be the value of A?

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Discrete Mathematics 4. NUMBER THEORY Lecture 7 Dr.-Ing. Erwin Sitompul