According to Wikipedia, “Number Theory is a branch of Pure Mathematics devoted primarily to the study of integers. Arithmetic - Arithmetic - Theory of divisors: At this point an interesting development occurs, for, so long as only additions and multiplications are performed with integers, the resulting numbers are invariably themselves integers—that is, numbers of the same kind as their antecedents. (b) aj1 if and only if a = 1. It is probably easier to recognize this as division by the algebraic re-arrangement: 1. n/k = q + r/k (0 ≤ r/k< 1) Euclid’s Algorithm. Definition. The division algorithm is basically just a fancy name for organizing a division problem in a nice equation. Not to be confused with Euclid's division lemma, Euclid's theorem, or Euclidean algorithm. Example. Thus \(z\) has a unique solution modulo \(n\),and division makes sense for this case. Divisibility. 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. 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). Prove that if $a,$ $b,$ and $c$ are integers with $a$ and $c$ nonzero, such that $a|b$ and $c|d,$ then $ac|bd.$, Exercise. We will use mathematical induction. Example. \[ z = x r + t n , k = z s - t y \] for all integers \(t\). Solution. Number Theory. $$ Notice $S$ is nonempty since $ab>a.$ By the Well-Ordering Axiom, $S$ must contain a least element, say $bk.$ Since $k\not= 0,$ there exists a natural number $q$ such that $k=q+1.$ Notice $b q\leq a$ since $bk$ is the least multiple of $b$ greater than $a.$ Thus there exists a natural number $r$ such that $a=bq+r.$ Notice $0\leq r.$ Assume, $r\geq b.$ Then there exists a natural number $m\geq 0$ such that $b+m=r.$ By substitution, $a=b(q+1)+m$ and so $bk=b(q+1)\leq a.$ This contradiction shows $r< b$ as needed. Extend the Division Algorithm by allowing negative divisors. The rules of sign division says that the quotient of two positive or two negative integers is a positive integer, while that of a negative integer and a positive integer is a negative integer. Lemma. Show that any integer of the form $6k+5$ is also of the form $3 j+2,$ but not conversely. From the previous statement, it is clear that every integer must have at least two divisors, namely 1 and the number itself. For if $a|n$ where $a$ and $n$ are positive integers, then $n=ak$ for some integer $k.$ Since $k$ is a positive integer, we see that $n=ak\geq a.$ Hence any nonzero integer $n$ can have at most $2|n|$ divisors. If $c|a$ and $c|b,$ then $c|(x a+y b)$ for any positive integers $x$ and $y.$. Let $b$ be an arbitrary natural number greater than $0$ and let $S$ be the set of multiples of $b$ that are greater than $a,$ namely, $$ S=\{b i \mid i\in \mathbb{N} \text{ and } bi>a\}. Lemma. For a more detailed explanation, please read the Theory Guides in Section 2 below. Division algorithms fall into two main categories: slow division and fast division. For example, when a number is divided by 7, the remainder after division will be an integer between 0 and 6. The division algorithm, therefore, is more or less an approach that guarantees that the long division process is actually foolproof. Add some text here. The Division Algorithm. These notes serve as course notes for an undergraduate course in number the-ory. Show that the square of every of odd integer is of the form $8k+1.$, Exercise. Prove if $a|b,$ then $a^n|b^n$ for any positive integer $n.$, Exercise. We say an integer $n$ is a linear combination of $a$ and $b$ if there exists integers $x$ and $y$ such that $n=ax+by.$ For example, $7$ is a linear combination of $3$ and $2$ since $7=2(2)+1(3).$. Edit. You will see many examples here. Number theory, Arithmetic. Similarly, dividing 954 by 8 and applying the division algorithm, we find 954=8\times 119+2 954 = 8×119+2 and hence we can conclude that the largest number before 954 which is a multiple of 8 is 954-2=952. [thm4] If a, b, c, m and n are integers, and if c ∣ a and c ∣ b, then c ∣ (ma + nb). The next lemma says that if an integer divides two other integers, then it divides any linear combination of these two integers. 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. 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. Discussion The division algorithm is probably one of the rst concepts you learned relative to the operation of division. That is, a = bq + r; 0 r < jbj. The algorithm that we present in this section is due to Euclid and has been known since ancient times. Addition, subtraction, and multiplication follow naturally from their integer counterparts, but we have complications with division. Theorem. First we prove existence. Exercise. History Talk (0) Share. Some are applied by hand, while others are employed by digital circuit designs and software. If a number $N$ is a factor of two number $s$ and $t$, then it is also a factor of the sum of and the difference between $s$ and $t$; and 4. Since $a|b$ certainly implies $a|b,$ the case for $k=1$ is trivial. An integer other than A number other than1is said to be aprimeif its only divisors are1and itself. Section 2.1 The Division Algorithm Subsection 2.1.1 Statement and examples. Lemma. Strictly speaking, it is not an algorithm. If $a,$ $b$ and $c\neq 0$ are integers, then $a|b$ if and only if $ac|bc.$, Exercise. Division algorithm. All rights reserved. If $a|b,$ then $a^n|b^n$ for any natural number $n.$. The first link in each item is to a Web page; the second is to a PDF file. (Transitive Property of Divisibility) Let $a,$ $b,$ and $c$ be integers. 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. $$ If $q_1=q_2$ then $r_1=r_2.$ Assume $q_1< q_2.$ Then $q_2=q_1+n$ for some natural number $n>0.$ This implies $$ 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. David Smith is the CEO and founder of Dave4Math. http://www.michael-penn.net In the book Elementary number theory by Jones a standard proof for division algorithm is provided. All 4 digit palindromic numbers are divisible by 11. In either case, $m(m+1)(m+2)$ must be even. Then there exist unique integers q and r so that a = bq + r and 0 r < jbj. The notes contain a useful introduction to important topics that need to be ad-dressed in a course in number theory. An algorithm describes a procedure for solving a problem. 0. We begin by defining how to perform basic arithmetic modulo \(n\), where \(n\) is a positive integer. 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. Prove that the fourth power of any integer is either of the form $5k$ or $5k+1.$, Exercise. Choose from 500 different sets of number theory flashcards on Quizlet. If a number $N$ is divisible by $m$, then it is also divisible by the factors of $m$; 2. 2. Prove that $5^n-2^n$ is divisible by $3$ for $n\geq 1.$, Exercise. Some number-theoretic problems that are yet unsolved are: 1. Certainly the sum, difference and product of any two integers is an integer. Prove that $7^n-1$ is divisible by $6$ for $n\geq 1.$, Exercise. The study of the integers is to a great extent the study of divisibility. Show $6$ divides the product of any three consecutive positive integers. The total number of times b was subtracted from a is the quotient, and the number r is the remainder. 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. 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 $a.$. Proof. In this video, we present a proof of the division algorithm and some examples of it in practice. Prove that, for each natural number $n,$ $7^n-2^n$ is divisible by $5.$. We call athe dividend, dthe divisor, qthe quotient, and r the remainder. Many lemmas exploring their basic properties are then proven. 2. So the number of trees marked with multiples of 8 is The same can not be said about the ratio of two integers. 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. Also, if it is possible to divide a number $m$, then it is equally possible to divide its negative. Theorem. 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. 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. (e) ajb and bja if and only if a = b. This preview shows page 1 - 3 out of 5 pages. Dave4Math » Number Theory » Divisibility (and the Division Algorithm). Proof. Proof. 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 $r$. Whence, $a^{k+1}|b^{k+1}$ as desired. For example, while 2 and 3 are integers, the ratio $2/3$ is not an integer. This characteristic changes drastically, however, as soon as division is introduced. 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. Show that $f_n\mid f_m$ when $n$ and $m$ are positive integers with $n\mid m.$, Exercise. Copyright © 2021 Dave4Math LLC. Lemma. The theorem does not tell us how to find the quotient and the remainder. [June 28, 2019] These notes were revised in Spring, 2019. Let $a$ and $b$ be integers. There are unique integers qand r, with 0 ≤r < d, such that a= dq+ r. Some mathematicians prefer to call it the division theorem. [Number Theory] Lecture 04 - Divisibility and the Division Algorithm.pdf - Math Elec 6 Number Theory Lecture 04 Divisibility and the Division Algorithm, 1 out of 1 people found this document helpful, Lecture 04 - Divisibility and the Division Algorithm, (2) Prove results involving divisibility of integers, (3) State, prove and apply the division algorithm, The following examples illustrate the concept of divisibility. A number of form 2 N has exactly N+1 divisors. Learn number theory with free interactive flashcards. De nition Let a and b be integers. We call q the quotient, r the remainder, and k the divisor. We assume a >0 in further slides! It abounds in problems that yet simple to state, are very hard to solve. 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. Need an assistance with a specific step of a specific Division Algorithm proof. 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. Number Theory is one of the oldest and most beautiful branches of Mathematics. The importance of the division algorithm is demonstrated through examples. (Linear Combinations) Let $a,$ $b,$ and $c$ be integers. Examples of … Exercise. Defining key concepts - ensure that you can explain the division algorithm Additional Learning To find out more about division, open the lesson titled Number Theory: Divisibility & Division Algorithm. 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. … Show that the product of every two integers of the form $6k+5$ is of the form $6k+1.$. (d) If ajb and bjc, then ajc. 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. With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. (Karl Friedrich Gauss) CSI2101 Discrete Structures Winter 2010: Intro to Number TheoryLucia Moura 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 have $$ x a+y b=x(m c)+y(n c)= c(x m+ y n) $$ Since $x m+ y n \in \mathbb{Z}$ we see that $c|(x a+y b)$ as desired. The notion of divisibility is motivated and defined. Examples demonstrating how to use the Division Algorithm as a method of proof are then given. Prove that the cube of any integer has one of the forms: $7k,$ $7k+1,$ $7k-1.$, Exercise. If $a|m$ and $a|(ms+nt)$ for some integers $a\neq 0,$ $m,$ $s,$ $n,$ and $t,$ then $a|nt.$, Exercise. Use the PDF if you want to print it. 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 < … The division algorithm describes what happens in long division. The division algorithm states that given an integer and a positive integer , there are unique integers and , with , for which . It is not actually an algorithm, but this is this theorem’s If $a | b$ and $b | c,$ then $a | c.$. \[ 1 = r y + s n\] Then the solutions for \(z, k\) are given by. About Dave and How He Can Help You. The Division Algorithm. Some computer languages use another de nition. Proof. Lemma. Let $a$ and $b$ be positive integers. (Division Algorithm) Given integers aand d, with d>0, there exists unique integers qand r, with 0 r