(3) Greatest Common Divisor
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Black Hills State University
Black Hills State University MATH 341 Concepts addressed: Mathematics Number sense and numeration: meaning and use of numbers; the standard algorithms of the four basic operations; appropriate computation strategies and reasonableness of results; methods of mathematical investigation; number patterns; place value; equivalence; factors and multiples; ratio, proportion, percent; representations; calculator strategies; and number lines Students will be able to demonstrate proficiency in utilizing the traditional algorithms for the four basic operations and to analyze non-traditional student-generated algorithms to perform these calculations. Students will be able to use non-decimal bases and a variety of numeration systems, such as Mayan, Roman, etc, to perform routine calculations. Students will be able to identify prime and composite numbers, compute the prime factorization of a composite number, determine whether a given number is prime or composite, and classify the counting numbers from 1 to 200 into prime or composite using the Sieve of Eratosthenes. A prime number is a number with exactly two factors. For example, 13 is a prime number because its only factors are 1 and itself. A composite number has more than two factors. For example, the number 6 is composite because its factors are 1,2,3, and 6. To determine if a larger number is prime, try all prime numbers less than the square root of the number. If any of those primes divide the original number, then the number is prime; otherwise, the number is composite. The number 1 is neither prime nor composite. It is not prime because it does not have two factors. It is not composite because, otherwise, it would nullify the Fundamental Theorem of Arithmetic, which states that every composite number has a unique prime factorization. -
Number Theory Learning Module 3 — the Greatest Common Divisor 1
Number Theory Learning Module 3 — The Greatest Common Divisor 1 1 Objectives. • Understand the definition of greatest common divisor (gcd). • Learn the basic the properties of the gcd. • Understand Euclid’s algorithm. • Learn basic proofing techniques for greatest common divisors. 2 The Greatest Common Divisor Classical Greek mathematics concerned itself mostly with geometry. The notion of measurement is fundamental to ge- ometry, and the Greeks were the first to provide a formal foundation for this concept. Surprisingly, however, they never used fractions to express measurements (and never developed an arithmetic of fractions). They expressed geometrical measurements as relations between ratios. In numerical terms, these are statements like: 168 is to 120 as 7 is to 4, (2.1) which we would write today as 168{120 7{5. Statements such as (2.1) were natural to greek mathematicians because they viewed measuring as the process of finding a “common integral measure”. For example, we have: 168 24 ¤ 7 120 24 ¤ 5; so that we can use the integer 24 as a “common unit” to measure the numbers 168 and 120. Going back to our example, notice that 24 is not the only common integral measure for the integers 168 and 120, since we also have, for example, 168 6 ¤ 28 and 120 6 ¤ 20. The number 24, however, is the largest integer that can be used to “measure” both 168 and 120, and gives the representation in lowest terms for their ratio. This motivates the following definition: Definition 2.1. Let a and b be integers that are not both zero. -
Primality Testing for Beginners
STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker http://dx.doi.org/10.1090/stml/070 Primality Testing for Beginners STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell Gerald B. Folland (Chair) Serge Tabachnikov The cover illustration is a variant of the Sieve of Eratosthenes (Sec- tion 1.5), showing the integers from 1 to 2704 colored by the number of their prime factors, including repeats. The illustration was created us- ing MATLAB. The back cover shows a phase plot of the Riemann zeta function (see Appendix A), which appears courtesy of Elias Wegert (www.visual.wegert.com). 2010 Mathematics Subject Classification. Primary 11-01, 11-02, 11Axx, 11Y11, 11Y16. For additional information and updates on this book, visit www.ams.org/bookpages/stml-70 Library of Congress Cataloging-in-Publication Data Rempe-Gillen, Lasse, 1978– author. [Primzahltests f¨ur Einsteiger. English] Primality testing for beginners / Lasse Rempe-Gillen, Rebecca Waldecker. pages cm. — (Student mathematical library ; volume 70) Translation of: Primzahltests f¨ur Einsteiger : Zahlentheorie - Algorithmik - Kryptographie. Includes bibliographical references and index. ISBN 978-0-8218-9883-3 (alk. paper) 1. Number theory. I. Waldecker, Rebecca, 1979– author. II. Title. QA241.R45813 2014 512.72—dc23 2013032423 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. -
1. the Euclidean Algorithm the Common Divisors of Two Numbers Are the Numbers That Are Divisors of Both of Them
1. The Euclidean Algorithm The common divisors of two numbers are the numbers that are divisors of both of them. For example, the divisors of 12 are 1, 2, 3, 4, 6, 12. The divisors of 18 are 1, 2, 3, 6, 9, 18. Thus, the common divisors of 12 and 18 are 1, 2, 3, 6. The greatest among these is, perhaps unsurprisingly, called the of 12 and 18. The usual mathematical notation for the greatest common divisor of two integers a and b are denoted by (a, b). Hence, (12, 18) = 6. The greatest common divisor is important for many reasons. For example, it can be used to calculate the of two numbers, i.e., the smallest positive integer that is a multiple of these numbers. The least common multiple of the numbers a and b can be calculated as ab . (a, b) For example, the least common multiple of 12 and 18 is 12 · 18 12 · 18 = . (12, 18) 6 Note that, in order to calculate the right-hand side here it would be counter- productive to multiply 12 and 18 together. It is much easier to do the calculation as follows: 12 · 18 12 = = 2 · 18 = 36. 6 6 · 18 That is, the least common multiple of 12 and 18 is 36. It is important to know the least common multiple when adding two fractions. For example, noting that 12 · 3 = 36 and 18 · 2 = 36, we have 5 7 5 · 3 7 · 2 15 14 15 + 14 29 + = + = + = = . 12 18 12 · 3 18 · 2 36 36 36 36 Note that this way of calculating the least common multiple works only for two numbers. -
The Pollard's Rho Method for Factoring Numbers
Foote 1 Corey Foote Dr. Thomas Kent Honors Number Theory Date: 11-30-11 The Pollard’s Rho Method for Factoring Numbers We are all familiar with the concepts of prime and composite numbers. We also know that a number is either prime or a product of primes. The Fundamental Theorem of Arithmetic states that every integer n ≥ 2 is either a prime or a product of primes, and the product is unique apart from the order in which the factors appear (Long, 55). The number 7, for example, is a prime number. It has only two factors, itself and 1. On the other hand 24 has a prime factorization of 2 3 × 3. Because its factors are not just 24 and 1, 24 is considered a composite number. The numbers 7 and 24 are easier to factor than larger numbers. We will look at the Sieve of Eratosthenes, an efficient factoring method for dealing with smaller numbers, followed by Pollard’s rho, a method that allows us how to factor large numbers into their primes. The Sieve of Eratosthenes allows us to find the prime numbers up to and including a particular number, n. First, we find the prime numbers that are less than or equal to √͢. Then we use these primes to see which of the numbers √͢ ≤ n - k, ..., n - 2, n - 1 ≤ n these primes properly divide. The remaining numbers are the prime numbers that are greater than √͢ and less than or equal to n. This method works because these prime numbers clearly cannot have any prime factor less than or equal to √͢, as the number would then be composite. -
RSA and Primality Testing
RSA and Primality Testing Joan Boyar, IMADA, University of Southern Denmark Studieretningsprojekter 2010 1 / 81 Outline Outline Symmetric key ■ Symmetric key cryptography Public key Number theory ■ RSA Public key cryptography RSA Modular ■ exponentiation Introduction to number theory RSA RSA ■ RSA Greatest common divisor Primality testing ■ Modular exponentiation Correctness of RSA Digital signatures ■ Greatest common divisor ■ Primality testing ■ Correctness of RSA ■ Digital signatures with RSA 2 / 81 Caesar cipher Outline Symmetric key Public key Number theory A B C D E F G H I J K L M N O RSA 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 RSA Modular D E F G H I J K L M N O P Q R exponentiation RSA 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 RSA Greatest common divisor Primality testing P Q R S T U V W X Y Z Æ Ø Å Correctness of RSA Digital signatures 15 16 17 18 19 20 21 22 23 24 25 26 27 28 S T U V W X Y Z Æ Ø Å A B C 18 19 20 21 22 23 24 25 26 27 28 0 1 2 E(m)= m + 3(mod 29) 3 / 81 Symmetric key systems Outline Suppose the following was encrypted using a Caesar cipher and the Symmetric key Public key Danish alphabet. The key is unknown. What does it say? Number theory RSA RSA Modular exponentiation RSA ZQOØQOØ, RI. RSA Greatest common divisor Primality testing Correctness of RSA Digital signatures 4 / 81 Symmetric key systems Outline Suppose the following was encrypted using a Caesar cipher and the Symmetric key Public key Danish alphabet. -
The Least Common Multiple of a Quadratic Sequence
The least common multiple of a quadratic sequence Javier Cilleruelo Abstract For any irreducible quadratic polynomial f(x) in Z[x] we obtain the estimate log l.c.m. (f(1); : : : ; f(n)) = n log n + Bn + o(n) where B is a constant depending on f. 1. Introduction The problem of estimating the least common multiple of the ¯rstPn positive integers was ¯rst investigated by Chebyshev [Che52] when he introduced the function ª(n) = pm6n log p = log l.c.m.(1; : : : ; n) in his study on the distribution of prime numbers. The prime number theorem asserts that ª(n) » n, so the asymptotic estimate log l.c.m.(1; : : : ; n) » n is equivalent to the prime number theorem. The analogous asymptotic estimate for any linear polynomial f(x) = ax + b is also known [Bat02] and it is a consequence of the prime number theorem for arithmetic progressions: q X 1 log l.c.m.(f(1); : : : ; f(n)) » n ; (1) Á(q) k 16k6q (k;q)=1 where q = a=(a; b). We address here the problem of estimating log l.c.m.(f(1); : : : ; f(n)) when f is an irreducible quadratic polynomial in Z[x]. When f is a reducible quadratic polynomial the asymptotic estimate is similar to that we obtain for linear polynomials. This case is studied in section x4 with considerably less e®ort than the irreducible case. We state our main theorem. Theorem 1. For any irreducible quadratic polynomial f(x) = ax2 + bx + c in Z[x] we have log l.c.m. -
Some Facts from Number Theory
Computer Science 52 Some Facts from Number Theory Fall Semester, 2014 These notes are adapted from a document that was prepared for a different course several years ago. They may be helpful as a summary of the facts that we use in Computer Science 52 as we study RSA encryption. The exercises in this document are left over from the previous course; although they are a tremendous amount of fun, they are not part of the assigned work for Computer Science 52. 1 Quotients and Remainders If m and n are integers, and n 6= 0, then there is a unique pair of integers q and r satisfying m = qn + r , and 0 ≤ r < jnj. Not surprisingly, q is the quotient and r is the remainder when m is divided by n.1 We say that n divides m, or n is a divisor of m, when the remainder is zero. The notation n j m is used to signify that n divides m. An integer p is a prime number if 1 < jpj and the only divisors of p are ±1 and ±p. We write a ≡ b (mod n) if b − a is divisible by n.2 This notion, called congruence, is an equivalence relation. All the usual laws of arithmetic hold for congruence; for example, we have the distributivity of multiplication over addition and the associa- tivity of multiplication. a(b + c) ≡ ab + ac (mod n) a(bc) ≡ (ab)c (mod n) When computing arithmetic results for a congruence, it does not hurt replace a value by its remainder on division by n. -
Eureka Math™ Tips for Parents Module 2
Grade 6 Eureka Math™ Tips for Parents Module 2 The chart below shows the relationships between various fractions and may be a great Key Words tool for your child throughout this module. Greatest Common Factor In this 19-lesson module, students complete The greatest common factor of two whole numbers (not both zero) is the their understanding of the four operations as they study division of whole numbers, division greatest whole number that is a by a fraction, division of decimals and factor of each number. For operations on multi-digit decimals. This example, the GCF of 24 and 36 is 12 expanded understanding serves to complete because when all of the factors of their study of the four operations with positive 24 and 36 are listed, the largest rational numbers, preparing students for factor they share is 12. understanding, locating, and ordering negative rational numbers and working with algebraic Least Common Multiple expressions. The least common multiple of two whole numbers is the least whole number greater than zero that is a What Came Before this Module: multiple of each number. For Below is an example of how a fraction bar model can be used to represent the Students added, subtracted, and example, the LCM of 4 and 6 is 12 quotient in a division problem. multiplied fractions and decimals (to because when the multiples of 4 and the hundredths place). They divided a 6 are listed, the smallest or first unit fraction by a non-zero whole multiple they share is 12. number as well as divided a whole number by a unit fraction. -
Divisibility and Greatest Common Divisors
DIVISIBILITY AND GREATEST COMMON DIVISORS KEITH CONRAD 1. Introduction We will begin with a review of divisibility among integers, mostly to set some notation and to indicate its properties. Then we will look at two important theorems involving greatest common divisors: Euclid's algorithm and Bezout's identity. The set of integers is denoted Z (from the German word Zahl = number). 2. The Divisibility Relation Definition 2.1. When a and b are integers, we say a divides b if b = ak for some k 2 Z. We then write a j b (read as \a divides b"). Example 2.2. We have 2 j 6 (because 6 = 2 · 3), 4 j (−12), and 5 j 0. We have ±1 j b for every b 2 Z. However, 6 does not divide 2 and 0 does not divide 5. Divisibility is a relation, much like inequalities. In particular, the relation 2 j 6 is not the number 3, even though 6 = 2 · 3. Such an error would be similar to the mistake of confusing the relation 5 < 9 with the number 9 − 5. Notice divisibility is not symmetric: if a j b, it is usually not true that b j a, so you should not confuse the roles of a and b in this relation: 4 j 20 but 20 - 4. Remark 2.3. Learn the definition of a j b as given in Definition 2.1, and not in the b 0 form \ a is an integer." It essentially amounts to the same thing (exception: 0 j 0 but 0 is not defined), however thinking about divisibility in terms of ratios will screw up your understanding of divisibility in other settings in algebra. -
Number Theory Section Summary: 2.2 the Greatest Common Divisor 1. Definitions 2. Theorems
Number Theory Section Summary: 2.2 The Greatest Common Divisor “Of special interest is the case in which the remainder in the Division Algo- rithm turns out to be zero.” (p. 20) 1. Definitions Divisible: An integer b is said to be divisible by an integer a =6 0, written a|b, if there exists some integer c such that b = ac. common divisor: An integer d is said to be a common divisor of a and b if both d|a and d|b. greatest common divisor: Let a and b be given integers, with at least one of them non-zero. The greatest common divisor of a and b, denoted gcd(a, b), is the positive integer d satisfying the following: (a) d|a and d|b (b) If c|a and c|b, then c ≤ d. relatively prime: Two integers a and b, not both zero, are said to be relatively prime whenever gcd(a, b)=1. 2. Theorems Theorem 2.2: For integers a, b, c, the following hold: (a) a|0, 1|a, a|a (b) a|1 if and only if a = ±1 (c) If a|b and c|d, then ac|bd. (d) If a|b and b|c, then a|c. (e) a|b and b|a if and only if a = ±b 1 (f) If a|b and b =6 0, then |a|≤|b|. (g) If a|b and a|c, then a|(bx + cy) for arbitrary integers x and y. Theorem 2.3: Given integers a and b, not both zero, there exists integers x and y such that gcd(a, b)= ax + by Proof: 2 Corollary: If a and b are given integers, not both zero, then the set T = {ax + by|x, y are integers} is precisely the set of all multiples of d = gcd(a, b). -
Euclidean Algorithm and Multiplicative Inverses
Fall 2019 Chris Christensen MAT/CSC 483 Finding Multiplicative Inverses Modulo n Two unequal numbers being set out, and the less begin continually subtracted in turn from the greater, if the number which is left never measures the one before it until an unit is left, the original numbers will be prime to one another. Euclid, The Elements, Book VII, Proposition 1, c. 300 BCE. It is not necessary to do trial and error to determine the multiplicative inverse of an integer modulo n. If the modulus being used is small (like 26) there are only a few possibilities to check (26); trial and error might be a good choice. However, some modern public key cryptosystems use very large moduli and require the determination of inverses. We will now examine a method (that is due to Euclid [c. 325 – 265 BCE]) that can be used to construct multiplicative inverses modulo n (when they exist). Euclid's Elements, in addition to geometry, contains a great deal of number theory – properties of the positive integers. The Euclidean algorithm is Propositions I - II of Book VII of Euclid’s Elements (and Propositions II – III of Book X). Euclid describes a process for determining the greatest common divisor (gcd) of two positive integers. 1 The Euclidean Algorithm to Find the Greatest Common Divisor Let us begin with the two positive integers, say, 13566 and 35742. Divide the smaller into the larger: 35742=×+ 2 13566 8610 Divide the remainder (8610) into the previous divisor (35742): 13566=×+ 1 8610 4956 Continue to divide remainders into previous divisors: 8610=×+ 1 4956 3654 4956=×+ 1 3654 1302 3654=×+ 2 1302 1050 1302=×+ 1 1050 252 1050=×+ 4 252 42 252= 6 × 42 The process stops when the remainder is 0.