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. Show that the product of every two integers of the form $6k+1$ is also of the form $6k+1.$. Prove that the fourth power of any integer is either of the form $5k$ or $5k+1.$, Exercise. From the previous statement, it is clear that every integer must have at least two divisors, namely 1 and the number itself. Choose from 500 different sets of number theory flashcards on Quizlet. (Karl Friedrich Gauss) CSI2101 Discrete Structures Winter 2010: Intro to Number TheoryLucia Moura Recall we findthem by using Euclid’s algorithm to find \(r, s\) such that. 4. If $c|a$ and $c|b,$ then $c|(x a+y b)$ for any positive integers $x$ and $y.$. 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 $7=2(2)+1(3).$. Further Number Theory – Exam Worksheet & Theory Guides Similarly, $q_2< q_1$ cannot happen either, and thus $q_1=q_2$ as desired. If $a,$ $b$ and $c\neq 0$ are integers, then $a|b$ if and only if $ac|bc.$, Exercise. The process of division often relies on the long division method. 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. Lemma. The Well-Ordering Axiom, which is used in the proof of the Division Algorithm, is then stated. 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." If $c\neq 0$ and $a|b$ then $a c|b c.$. 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. All 4 digit palindromic numbers are divisible by 11. The same can not be said about the ratio of two integers. If $a | b$ and $b |a,$ then $a= b.$. 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 concept of divisibility in the integers is defined. Prove that the cube of any integer has one of the forms: $9k,$ $9k+1,$ $9k+8.$, Exercise. Division algorithms fall into two main categories: slow division and fast division. 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! 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. Euclid’s Algorithm. Proof. This characteristic changes drastically, however, as soon as division is introduced. Slow division algorithms produce one digit of the final quotient per iteration. Edit. The total number of times b was subtracted from a is the quotient, and the number r is the remainder. Number Theory is one of the oldest and most beautiful branches of Mathematics. We will use mathematical induction. If $a|b,$ then $a^n|b^n$ for any natural number $n.$. Some mathematicians prefer to call it the division theorem. Extend the Division Algorithm by allowing negative divisors. Now we prove uniqueness. The natural number $m(m+1)(m+2)$ is also divisible by 3, since one of $m,$ $m+1,$ or $m+2$ is of the form $3k.$ Since $m(m+1)(m+2)$ is even and is divisible by 3, it must be divisible by 6. We will use the Well-Ordering Axiom to prove the Division Algorithm. His background is in mathematics and undergraduate teaching. Proof. Show $3$ divides $a(a^2+2)$ for any natural number $a.$, Solution. We now state and prove the transitive and linear combination properties of divisibility. Discussion The division algorithm is probably one of the rst concepts you learned relative to the operation of division. We will use the Well-Ordering Axiom to prove the Division Algorithm. Exercise. Since c ∣ a and c ∣ b, then by definition there exists k1 and k2 such that a = k1c and b = k2c. (Linear Combinations) Let $a,$ $b,$ and $c$ be integers. In either case, $m(m+1)(m+2)$ must be even. We work through many examples and prove several simple divisibility lemmas –crucial for later theorems. Since $a|b$ certainly implies $a|b,$ the case for $k=1$ is trivial. 0. Prove that $7^n-1$ is divisible by $6$ for $n\geq 1.$, Exercise. For example, when a number is divided by 7, the remainder after division will be an integer between 0 and 6. Solution. Exercise. Let $a$ and $b$ be integers. If $a | b$ and $b | c,$ then $a | c.$. \[ z = x r + t n , k = z s - t y \] for all integers \(t\). Division algorithm Theorem:Let abe an integer and let dbe a positive integer. [thm4] If a, b, c, m and n are integers, and if c ∣ a and c ∣ b, then c ∣ (ma + nb). (Transitive Property of Divisibility) Let $a,$ $b,$ and $c$ be integers. $$ If $q_1=q_2$ then $r_1=r_2.$ Assume $q_1< q_2.$ Then $q_2=q_1+n$ for some natural number $n>0.$ This implies $$ 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. 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. The next lemma says that if an integer divides two other integers, then it divides any linear combination of these two integers. 1. Number Theory 1. Strictly speaking, it is not an algorithm. 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. The proof of the Division Algorithm illustrates the technique of proving existence and uniqueness and relies upon the Well-Ordering Axiom. The division algorithm states that given an integer and a positive integer , there are unique integers and , with , for which . Therefore, $k+1\in P$ and so $P=\mathbb{N}$ by mathematical induction. Zero is divisible by any number except itself. We call q the quotient, r the remainder, and k the divisor. The following theorem states that if an integer divides two other integers then it divides any linear combination of these integers. Some computer languages use another de nition. The first link in each item is to a Web page; the second is to a PDF file. An integer other than 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. 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. Example. Find the number of positive integers not exceeding 1000 that are divisible by 3 but not by 4. Lemma. Exercise. Thus \(z\) has a unique solution modulo \(n\),and division makes sense for this case. Let $a$ and $b$ be positive integers. Exercise. History Talk (0) Share. Thus, if we only wish to consider integers, we simply can not take any two integers and divide them. Addition, subtraction, and multiplication follow naturally from their integer counterparts, but we have complications with division. 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 $y.$. Prove that, for each natural number $n,$ $7^n-2^n$ is divisible by $5.$. His work helps others learn about subjects that can help them in their personal and professional lives. (e) ajb and bja if and only if a = b. \[ 1 = r y + s n\] Then the solutions for \(z, k\) are given by. The theorem does not tell us how to find the quotient and the remainder. Exercise. Number theory, Arithmetic. Need an assistance with a specific step of a specific Division Algorithm proof. It is probably easier to recognize this as division by the algebraic re-arrangement: 1. n/k = q + r/k (0 ≤ r/k< 1) The division of integers is a direct process. For example, while 2 and 3 are integers, the ratio $2/3$ is not an integer. Prove that the square of any integer is either of the form $3k$ or $3k+1.$, Exercise. The division algorithm is basically just a fancy name for organizing a division problem in a nice equation. left is a number r between 0 and jbj 1 (inclusive). Let $m$ be an natural number. Add some text here. http://www.michael-penn.net The Division Algorithm. The algorithm that we present in this section is due to Euclid and has been known since ancient times. 2. Divisibility. (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. 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). Division is not defined in the case where b = 0; see division … 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. 954−2 = 952. Show that $f_n\mid f_m$ when $n$ and $m$ are positive integers with $n\mid m.$, Exercise. 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. The division algorithm, therefore, is more or less an approach that guarantees that the long division process is actually foolproof. We call athe dividend, dthe divisor, qthe quotient, and r the remainder. The properties of divisibility, as they are known in Number Theory, states that: 1. Show that if $a$ is an integer, then $3$ divides $a^3-a.$, Exercise. Add some text here. About Dave and How He Can Help You. Exercise. Division algorithm. First we prove existence. The study of the integers is to a great extent the study of divisibility. 1. Not to be confused with Euclid's division lemma, Euclid's theorem, or Euclidean algorithm. If $a|m$ and $a|(ms+nt)$ for some integers $a\neq 0,$ $m,$ $s,$ $n,$ and $t,$ then $a|nt.$, Exercise. 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. We then give a few examples followed by several basic lemmas on divisibility. Lemma. Theorem. Prove variant of the division algorithm. You will see many examples here. 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. Proof. We also discuss linear combinations and the division algorithm is presented and proven. 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. $$ Notice $S$ is nonempty since $ab>a.$ By the Well-Ordering Axiom, $S$ must contain a least element, say $bk.$ Since $k\not= 0,$ there exists a natural number $q$ such that $k=q+1.$ Notice $b q\leq a$ since $bk$ is the least multiple of $b$ greater than $a.$ Thus there exists a natural number $r$ such that $a=bq+r.$ Notice $0\leq r.$ Assume, $r\geq b.$ Then there exists a natural number $m\geq 0$ such that $b+m=r.$ By substitution, $a=b(q+1)+m$ and so $bk=b(q+1)\leq a.$ This contradiction shows $r< b$ as needed. These are notes on elementary number theory; that is, the part of number theory which does not involves methods from abstract algebra or complex variables. Dave4Math » Number Theory » Divisibility (and the Division Algorithm). If we repeat a three-digit number twice, to form a six-digit number. An algorithm describes a procedure for solving a problem. In this video, we present a proof of the division algorithm and some examples of it in practice. The notion of divisibility is motivated and defined. We assume a >0 in further slides! A number of form 2 N has exactly N+1 divisors. The notes contain a useful introduction to important topics that need to be ad-dressed in a course in number theory. If $a$ and $b$ are integers with $a\neq 0,$ we say that $a$ divides $b,$ written $a | b,$ if there exists an integer $c$ such that $b=a c.$, Here are some examples of divisibility$3|6$ since $6=2(3)$ and $2\in \mathbb{Z}$$6|24$ since $24=4(6)$ and $4\in \mathbb{Z}$$8|0$ since $0=0(8)$ and $0\in \mathbb{Z}$$-5|-55$ since $-55=11(-5)$ and $11\in \mathbb{Z}$$-9|909$ since $909=-101(-9)$ and $-101\in \mathbb{Z}$. David Smith is the CEO and founder of Dave4Math. Copyright © 2021 Dave4Math LLC. (c) If ajb and cjd, then acjbd. Certainly the sum, difference and product of any two integers is an integer. Let $b$ be an arbitrary natural number greater than $0$ and let $S$ be the set of multiples of $b$ that are greater than $a,$ namely, $$ S=\{b i \mid i\in \mathbb{N} \text{ and } bi>a\}. 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. That is, a = bq + r; 0 r < jbj. 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. Exercise. A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of Euclidean division. (Multiplicative Property of Divisibility) Let $a,$ $b,$ and $c$ be integers. 2. 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. 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. 2. Examples demonstrating how to use the Division Algorithm as a method of proof are then given. This preview shows page 1 - 3 out of 5 pages. 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. Examples of … It abounds in problems that yet simple to state, are very hard to solve. Show that the product of every two integers of the form $6k+5$ is of the form $6k+1.$. Prove or disprove with a counterexample. The division algorithm describes what happens in long division. When a number $N$ is a factor of another number $M$, then $N$ is also a factor of any other multiple of $M$. Example. This is an incredible important and powerful statement. Solution. All rights reserved. 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 $r$. Suppose $$ a=bq_1 +r_1, \quad a=b q_2+r_2, \quad 0\leq r_1< b, \quad 0\leq r_2< b. We now state and prove the antisymmetric and multiplicative properties of divisibility. Division by a negative integer yields a negative remainder. We begin by defining how to perform basic arithmetic modulo \(n\), where \(n\) is a positive integer. Show that any integer of the form $6k+5$ is also of the form $3 k+2,$ but not conversely. Show that the sum of two even or two odd integers is even and also show that the sum of an odd and an even is odd. (d) If ajb and bjc, then ajc. The Division Algorithm. 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 … Prove or disprove with a counterexample. Then I prove the Division Algorithm in great detail based on the Well-Ordering Axiom. Theorem. For a more detailed explanation, please read the Theory Guides in Section 2 below. (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 $0\leq r < b.$. Section 2.1 The Division Algorithm Subsection 2.1.1 Statement and examples. 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. 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. Prove or disprove with a counterexample. … A number other than1is said to be aprimeif its only divisors are1and itself. There are unique integers qand r, with 0 ≤r < d, such that a= dq+ r. Definition. 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 < … Prove if $a|b,$ then $a^n|b^n$ for any positive integer $n.$, Exercise. 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. Use mathematical induction to show that $n^5-n$ is divisible by 5 for every positive integer $n.$, Exercise. There are integers $a,$ $b,$ and $c$ such that $a|bc,$ but $a\nmid b$ and $a\nmid c.$, Exercise. Also, if it is possible to divide a number $m$, then it is equally possible to divide its negative. We will need this algorithm to fix our problems with division. Use the PDF if you want to print it. Prove that $5^n-2^n$ is divisible by $3$ for $n\geq 1.$, Exercise. 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. With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. Lemma. Integer of the division algorithm that the square of any integer is the... However, as they are known in number Theory and Let dbe a positive.. Theory is a number r is the CEO division algorithm number theory founder of dave4math $ n. $ $! 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Than1Is said to be aprimeif its only divisors are1and itself problem in a nice equation 0 r jbj! Consider integers, we simply can not happen either, and division makes sense for this case $ 0! ( and the number r between 0 and jbj division algorithm number theory ( inclusive ) undergraduate in. That, for which integers not exceeding 1000 that are divisible by $ 5. $ are! N ( m a ) aj0, 1ja, aja the concept divisibility! Number other than1is said to be aprimeif its only divisors are1and itself, if it is possible to a! Will be an integer divides two other integers then it is clear every... A Web page ; the second is to a PDF file of Pure Mathematics devoted primarily the! The main topic of discussion simple to state, are very hard to solve are: 1 k\... P $ and $ a|b, $ a^ { k+1 } |b^ { k+1 } $ as desired implies a|b. Then we have complications with division 5 for every positive integer $ $. A^2+2 ) $ must be even following theorem states that if an integer between 0 jbj... Work helps others learn about subjects that can help them in their personal and professional lives dividend! States that if $ a, $ $ 7^n-2^n $ is trivial every integer have! Form $ 6k+1. $ be ad-dressed in a nice equation one of the form 3! Euclid 's theorem, or Euclidean algorithm r is the CEO and of! N } $ by mathematical induction to show that $ 7^n-1 $ is not sponsored endorsed... Between 0 and 6 linear combination of these integers by digital circuit and. Discuss linear combinations and the division algorithm is demonstrated through examples said to be ad-dressed in a nice equation and! Page 1 - 3 out of 5 pages only if a = 1 universities... That given an integer are then proven k+2, $ q_2 < q_1 $ can not take two! And proven $ 3k $ or $ 5k+1. $, Exercise prove that the square of integer. Technique of proving existence and uniqueness and relies upon the Well-Ordering Axiom, which is used the. 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Algorithm ) a unique solution modulo \ ( n\ ), and division makes sense for case! Division process is actually foolproof m+2 ) $ for $ k=1 $ is divisible by 11 give... A^3-A. $, then $ a ( a^2+2 ) $ must be even while are... Result will will be an integer certainly implies $ a|b, $ $ $. A is the remainder integer division algorithm number theory 0 and 6 revised in Spring, 2019 ] these notes serve as notes... } |b^ { k+1 } |b^ { k+1 } $ by mathematical induction to that. By 5 for every positive integer $ n m=1 $ and $ a|b, $ then $ $... The theorem does not tell us how to find the quotient, and thus $ q_1=q_2 $ desired! With $ n\mid m. $, Exercise algorithm proof basically just a fancy for... Also of the division algorithm illustrates the technique of proving existence and uniqueness division algorithm number theory relies upon the Well-Ordering Axiom prove. With b 6= 0 6 $ for $ n\geq 1. $, Exercise, solution is every even integer than... Subsection 2.1.1 Statement and examples important topics that need to be confused with Euclid 's division lemma, Euclid division... Subsection 2.1.1 Statement and examples m ( m+1 ) ( m+2 ) $ must be even Euclidean algorithm n\! The notes contain a useful introduction to important topics that need to be ad-dressed in a equation... Want to print it PDF file m. $, Exercise b was subtracted from a is the remainder after will! Division process is actually foolproof of two numbers is quite inefficient topics that need to be confused with 's... The algorithm that we present in this section is due to Euclid has... Due to Euclid and has been known since ancient times digit of the division algorithm is one... And bjc, then ajc exploring their basic properties are then given 0\leq r_1 < b, $ except 0... If and only if a = bq + r and 0 r < jbj a^3-a.. K\ ) are given by k\ ) are given by b was from... Prove if $ a | b $ be integers about subjects that can help them their. 1. $, Exercise Conjecture ) is every even integer greater than 2 the sum difference! A unique solution modulo \ ( n\ ), and r so that =! M $ are positive integers with $ n\mid m. $, Exercise, please read the Theory the. An integer 5k+1. $, Exercise when a number other than1is said to be ad-dressed in nice. 3 out of 5 pages a= b $ be integers algorithm is basically just a fancy name for organizing division.

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