Thus \(z\) has a unique solution modulo \(n\),and division makes sense for this case. We assume a >0 in further slides! 2. … 1. (e) ajb and bja if and only if a = b. (c) If ajb and cjd, then acjbd. Number Theory: Part 3 1 The Euclidean Algorithm We begin this lecture by introducing of a very famous and historical “ algorithm” for finding the greatest common divisor of two numbers. It is probably easier to recognize this as division by the algebraic re-arrangement: 1. n/k = q + r/k (0 ≤ r/k< 1) Let's start off with the division algorithm. Suppose $c|a$ and $c|b.$ Then there exists integers $m$ and $n$ such that $a=m c$ and $b=n c.$ Assume $x$ and $y$ are arbitrary integers. For example, when a number is divided by 7, the remainder after division will be an integer between 0 and 6. The result will will be divisible by 7, 11 and 13, and dividing by all three will give your original three-digit number. Show that the product of every two integers of the form $6k+1$ is also of the form $6k+1.$. In number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely: Euclid's lemma — If a prime p divides the product ab of two integers a and b, then p must divide at least one of those integers a and b. In addition to showing the divisibility relationship between any two non zero integers, it is worth noting that such relationships are characterized by certain properties. Theorem. From the previous statement, it is clear that every integer must have at least two divisors, namely 1 and the number itself. Division algorithms fall into two main categories: slow division and fast division. Theorem 5.2.1The Division Algorithm Let a;b 2Z, with b 6= 0 . For example, while 2 and 3 are integers, the ratio $2/3$ is not an integer. Show that any integer of the form $6k+5$ is also of the form $3 k+2,$ but not conversely. We need to show that $m(m+1)(m+2)$ is of the form $6 k.$ The division algorithm yields that $m$ is either even or odd. (d) If ajb and bjc, then ajc. If $c|a$ and $c|b,$ then $c|(x a+y b)$ for any positive integers $x$ and $y.$. Division algorithm Theorem:Let abe an integer and let dbe a positive integer. (Multiplicative Property of Divisibility) Let $a,$ $b,$ and $c$ be integers. (Division Algorithm) If $a$ and $b$ are nonzero positive integers, then there are unique positive integers $q$ and $r$ such that $a=bq+r$ where $0\leq r < b.$ Proof. There are integers $a,$ $b,$ and $c$ such that $a|bc,$ but $a\nmid b$ and $a\nmid c.$, Exercise. The Division Algorithm. Discussion The division algorithm is probably one of the rst concepts you learned relative to the operation of division. His work helps others learn about subjects that can help them in their personal and professional lives. left is a number r between 0 and jbj 1 (inclusive). The process of division often relies on the long division method. Most if not all universities worldwide offer introductory courses in number theory for math majors and in many cases as an elective course. Some are applied by hand, while others are employed by digital circuit designs and software. Cebu Technological University (formerly Cebu State College of Science and Technology), [Number Theory] Lecture 03 - Induction and Pigeonhole Principles.pdf, [Number Theory] Lecture 02 - Some Important Notations.pdf, [Number Theory] Lecture 01 - The Number System.pdf, Cebu Technological University (formerly Cebu State College of Science and Technology) • MATH-C 221, Cebu Technological University (formerly Cebu State College of Science and Technology) • EDU 227, [Number Theory] Lecture 06 - GCDs, LCMs, and the Euclidean Algorithm.pdf, [Number Theory] Lecture 07 - The Fudamental Theorem of Arithmetic.pdf, Cebu Technological University (formerly Cebu State College of Science and Technology) • COE 101. Use mathematical induction to show that $n^5-n$ is divisible by 5 for every positive integer $n.$, Exercise. Some computer languages use another de nition. Zero is divisible by any number except itself. Example. Number Theory is one of the oldest and most beautiful branches of Mathematics. $$ Notice $S$ is nonempty since $ab>a.$ By the Well-Ordering Axiom, $S$ must contain a least element, say $bk.$ Since $k\not= 0,$ there exists a natural number $q$ such that $k=q+1.$ Notice $b q\leq a$ since $bk$ is the least multiple of $b$ greater than $a.$ Thus there exists a natural number $r$ such that $a=bq+r.$ Notice $0\leq r.$ Assume, $r\geq b.$ Then there exists a natural number $m\geq 0$ such that $b+m=r.$ By substitution, $a=b(q+1)+m$ and so $bk=b(q+1)\leq a.$ This contradiction shows $r< b$ as needed. Not to be confused with Euclid's division lemma, Euclid's theorem, or Euclidean algorithm. The importance of the division algorithm is demonstrated through examples. In the book Elementary number theory by Jones a standard proof for division algorithm is provided. Show that any integer of the form $6k+5$ is also of the form $3 j+2,$ but not conversely. (b) aj1 if and only if a = 1. Recall we findthem by using Euclid’s algorithm to find \(r, s\) such that. We call athe dividend, dthe divisor, qthe quotient, and r the remainder. Assume that $a^k|b^k$ holds for some natural number $k>1.$ Then there exists an integer $m$ such that $b^k=m a^k.$ Then \begin{align*} b^{k+1} & =b b^k =b \left(m a^k\right) \\ & =(b m )a^k =(m’ a m )a^k =M a^{k+1} \end{align*} where $m’$ and $M$ are integers. Let $P$ be the set of natural number for which $7^n-2^n$ is divisible by $5.$ Clearly, $7^1-2^1=5$ is divisible by $5,$ so $P$ is nonempty with $0\in P.$ Assume $k\in P.$ We find \begin{align*} 7^{k+1}-2^{k+1} & = 7\cdot 7^k-2\cdot 2^k \\ & = 7\cdot 7^k-7\cdot 2^k+7\cdot 2^k-2\cdot 2^k \\ & = 7(7^k- 2^k)+2^k(7 -2) \end{align*} The induction hypothesis is that $(7^k- 2^k)$ is divisible by 5. The proof of the Division Algorithm illustrates the technique of proving existence and uniqueness and relies upon the Well-Ordering Axiom. The first link in each item is to a Web page; the second is to a PDF file. Math Elec 6 Number Theory Lecture 04 - Divisibility and the Division Algorithm Julius D. Selle Lecture Objectives (1) Define divisibility (2) Prove results involving divisibility of integers (3) State, prove and apply the division algorithm Experts summarize Number Theory as the study of primes. Its handiness draws from the fact that it not only makes the process of division easier, but also in its use in finding the proof of the Fundamental Theory of Arithmetic. We now state and prove the antisymmetric and multiplicative properties of divisibility. Proof. The following theorem states that if an integer divides two other integers then it divides any linear combination of these integers. A number other than1is said to be aprimeif its only divisors are1and itself. The concept of divisibility in the integers is defined. Prove that the square of any integer is either of the form $3k$ or $3k+1.$, Exercise. Let $a$ and $b$ be positive integers. When a number $N$ is a factor of another number $M$, then $N$ is also a factor of any other multiple of $M$. We will use mathematical induction. Course Hero is not sponsored or endorsed by any college or university. 4. Edit. For any positive integer a and integer b, there exist unique integers q and r such that b = qa + r and 0 ≤ r < a, with r = 0 iﬀ a | b. Add some text here. Specifically, prove that whenever $a$ and $b\neq 0$ are integers, there are unique integers $q$ and $r$ such that $a=bq+r,$ where $0\leq r < |b|.$, Exercise. (Division Algorithm) Given integers aand d, with d>0, there exists unique integers qand r, with 0 r 0, there exist unique integers q and r satisfying a = qb+ r 0 r < b. Many lemmas exploring their basic properties are then proven. For a more detailed explanation, please read the Theory Guides in Section 2 below. Let $b$ be an arbitrary natural number greater than $0$ and let $S$ be the set of multiples of $b$ that are greater than $a,$ namely, $$ S=\{b i \mid i\in \mathbb{N} \text{ and } bi>a\}. The advantage of the Division Algorithm is that it allows us to prove statements about the positive integers (integers) by considering only a finite number of cases. About Dave and How He Can Help You. The proof of the Division Algorithm illustrates the technique of proving existence and uniqueness and relies upon the Well-Ordering Axiom. Prove or disprove with a counterexample. Since $a|b$ certainly implies $a|b,$ the case for $k=1$ is trivial. Exercise. The theorem is frequently referred to as the division algorithm (although it is a theorem and not an algorithm), because its proof as given below lends itself to a simple division algorithm for computing q and r (see the section Proof for more). An integer other than Proof. The total number of times b was subtracted from a is the quotient, and the number r is the remainder. Prove variant of the division algorithm. (Karl Friedrich Gauss) CSI2101 Discrete Structures Winter 2010: Intro to Number TheoryLucia Moura Addition, subtraction, and multiplication follow naturally from their integer counterparts, but we have complications with division. Examples demonstrating how to use the Division Algorithm as a method of proof are then given. If $a|b,$ then $a^n|b^n$ for any natural number $n.$. These notes serve as course notes for an undergraduate course in number the-ory. (Division Algorithm) If $a$ and $b$ are nonzero positive integers, then there are unique positive integers $q$ and $r$ such that $a=bq+r$ where $0\leq r < b.$. Prove that if $a$ ad $b$ are integers, with $b>0,$ then there exists unique integers $q$ and $r$ satisfying $a=bq+r,$ where $2b\leq r < 3b.$, Exercise. In number theory, we study about integers, rational and irrational, prime numbers etc and some number system related concepts like Fermat theorem, Wilson’s theorem, Euclid’s algorithm etc. Lemma. Then we have $$ a=n b= n(m a) = (n m) a. We will use the Well-Ordering Axiom to prove the Division Algorithm. Number theory, Arithmetic. \[ 1 = r y + s n\] Then the solutions for \(z, k\) are given by. If $a | b$ and $b | c,$ then $a | c.$. Let $m$ be an natural number. Suppose $a|b.$ Then there exists an integer $n$ such that $b=n a.$ By substitution we find, $$ b c=(n c) a=(a c) n. $$ Since $c\neq 0,$ it follows that $ac\neq 0,$ and so $a c| b c$ as needed. The same can not be said about the ratio of two integers. The rules of sign division says that the quotient of two positive or two negative integers is a positive integer, while that of a negative integer and a positive integer is a negative integer. This is an incredible important and powerful statement. The division of integers is a direct process. Then there exist unique integers q and r so that a = bq + r and 0 r < jbj. We now state and prove the transitive and linear combination properties of divisibility. The Integers and Division Primes and Greatest Common Divisor Applications Introduction to Number Theory and its Applications Lucia Moura Winter 2010 \Mathematics is the queen of sciences and the theory of numbers is the queen of mathematics." The notes contain a useful introduction to important topics that need to be ad-dressed in a course in number theory. We begin by stating the definition of divisibility, the main topic of discussion. Lemma. All 4 digit palindromic numbers are divisible by 11. Thus, if we only wish to consider integers, we simply can not take any two integers and divide them. According to Wikipedia, “Number Theory is a branch of Pure Mathematics devoted primarily to the study of integers. Some mathematicians prefer to call it the division theorem. Similarly, $q_2< q_1$ cannot happen either, and thus $q_1=q_2$ as desired. We say an integer $a$ is of the form $bq+r$ if there exists integers $b,$ $q,$ and $r$ such that $a=bq+r.$ Notice that the division algorithm, in a certain sense, measures the divisibility of $a$ by $b$ using a remainder $r$. In this video, we present a proof of the division algorithm and some examples of it in practice. Exercise. This is the familiar elementary school fact that if you divide an integer \(a\) by a positive integer \(b\text{,}\) you will always get an integer … For integers a,b,c,d, the following hold: (a) aj0, 1ja, aja. The division algorithm states that given an integer and a positive integer , there are unique integers and , with , for which . The division algorithm is basically just a fancy name for organizing a division problem in a nice equation. If $a,$ $b$ and $c\neq 0$ are integers, then $a|b$ if and only if $ac|bc.$, Exercise. For signed integers, the easiest and most preferred approach is to operate with their absolute values, and then apply the rules of sign division to determine the applicable sign. Section 2.1 The Division Algorithm Subsection 2.1.1 Statement and examples. Before we state and prove the Division Algorithm, let’s recall the Well-Ordering Axiom, namely: Every nonempty set of positive integers contains a least element. 5 mod3 =5 3 b5 =3 c=2 5 mod 3 =5 ( 3 )b5 =( 3 )c= 1 5 mod3 = 5 3 b 5 =3 c=1 5 mod 3 = 5 ( 3 )b 5 =( 3 )c= 2 Be careful! Definition. We also discuss linear combinations and the division algorithm is presented and proven. Prove or disprove with a counterexample. 2. For if $a|n$ where $a$ and $n$ are positive integers, then $n=ak$ for some integer $k.$ Since $k$ is a positive integer, we see that $n=ak\geq a.$ Hence any nonzero integer $n$ can have at most $2|n|$ divisors. Divisibility and the Euclidean Algorithm Deﬁnition 2.1For integers a and b, b 6= 0, b is called adivisorof a, if there exists an integer c such that a=bc. $$ If $q_1=q_2$ then $r_1=r_2.$ Assume $q_1< q_2.$ Then $q_2=q_1+n$ for some natural number $n>0.$ This implies $$ r_1=a-b q_1=bq_2+r_2-b q_1=b n +r_2\geq b n\geq b $$ which is contrary to $r_1< b.$ Thus $q_1< q_2$ cannot happen. (Antisymmetric Property of Divisibility) Let $a$ and $b$ be nonzero positive integers. Solution. Just for context here is Theorem 1.1: If $a$ and $b$ are integers with $b > 0$, then there is a unique pair of integers $q$ and $r$ such that $$a=qb+r$$ and $$0\le r < … If a number $N$ is divisible by both $p$ and $q$, where $p$ and $q$ are co-prime numbers, then $N$ is also divisible by the product of $p$ and $q$; 3. Divisibility. Now we prove uniqueness. Learn number theory with free interactive flashcards. Exercise. The properties of divisibility, as they are known in Number Theory, states that: 1. The Well-Ordering Axiom, which is used in the proof of the Division Algorithm, is then stated. Given nonzero integers $a, b,$ and $c$ show that $a|b$ and $a|c$ implies $a|(b x+c y)$ for any integers $x$ and $y.$. \[ z = x r + t n , k = z s - t y \] for all integers \(t\). Prove that, for each natural number $n,$ $7^n-2^n$ is divisible by $5.$. We will need this algorithm to fix our problems with division. If a number $N$ is divisible by $m$, then it is also divisible by the factors of $m$; 2. Division is not defined in the case where b = 0; see division … Since c ∣ a and c ∣ b, then by definition there exists k1 and k2 such that a = k1c and b = k2c. 1. Show $6$ divides the product of any three consecutive positive integers. So the number of trees marked with multiples of 8 is Proof. For any integer n and any k > 0, there is a unique q and rsuch that: 1. n = qk + r (with 0 ≤ r < k) Here n is known as dividend. http://www.michael-penn.net Prove that $7^n-1$ is divisible by $6$ for $n\geq 1.$, Exercise. Proof. Whence, $a^{k+1}|b^{k+1}$ as desired. $$ Thus, $n m=1$ and so in particular $n= 1.$ Whence, $a= b$ as desired. You will see many examples here. The division algorithm describes what happens in long division. Then I prove the Division Algorithm in great detail based on the Well-Ordering Axiom. With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. There are unique integers qand r, with 0 ≤r < d, such that a= dq+ r. Examples. Show that the product of two odd integers is odd and also show that the product of two integers is even if either or one of them is even. Some number-theoretic problems that are yet unsolved are: 1. Lemma. Exercise. Find the number of positive integers not exceeding 1000 that are divisible by 3 but not by 4. Exercise. We then give a few examples followed by several basic lemmas on divisibility. Prove that the cube of any integer has one of the forms: $7k,$ $7k+1,$ $7k-1.$, Exercise. Show that if $a$ and $b$ are positive integers and $a|b,$ then $a\leq b.$, Exercise. It also follows that if it is possible to divide two numbers $m$ and $n$ individually, then it is also possible to divide their sum. We begin by defining how to perform basic arithmetic modulo \(n\), where \(n\) is a positive integer. Equivalently, we need to show that $a\left(a^2+2\right)$ is of the form $3k$ for some $k$ for any natural number $a.$ By the division algorithm, $a$ has exactly one of the forms $3 k,$ $3k+1,$ or $3k+2.$ If $a=3k+1$ for some $k,$ then $$ (3k+1)\left((3k+1)^2+2\right)=3(3k+1)\left(3k^2+2k+1\right) $$ which shows $3|a(a^2+2).$ If $a=3k+2$ for some $k,$ then $$ (3k+2) \left( (3k+2)^2+2\right)=3(3k+2)\left(3k^2+4k+2\right) $$ which shows $3|a(a^2+2).$ Finally, if $a$ is of the form $3k$ then we have $$ a \left(a^2+2\right) =3k\left(9k^2+2\right) $$ which shows $3|a(a^2+2).$ Therefore, in all possible cases, $3|a(a^2+2))$ for any positive natural number $a.$. We call q the quotient, r the remainder, and k the divisor. If $a|m$ and $a|(ms+nt)$ for some integers $a\neq 0,$ $m,$ $s,$ $n,$ and $t,$ then $a|nt.$, Exercise. 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