Prove that $5^n-2^n$ is divisible by $3$ for $n\geq 1.$, Exercise. Extend the Division Algorithm by allowing negative divisors. 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. 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 assume a >0 in further slides! His background is in mathematics and undergraduate teaching. Some are applied by hand, while others are employed by digital circuit designs and software. (Karl Friedrich Gauss) CSI2101 Discrete Structures Winter 2010: Intro to Number TheoryLucia Moura 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. Exercise. The concept of divisibility in the integers is defined. The total number of times b was subtracted from a is the quotient, and the number r is the remainder. Prove or disprove with a counterexample. Whence, $a^{k+1}|b^{k+1}$ as desired. Addition, subtraction, and multiplication follow naturally from their integer counterparts, but we have complications with division. 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. Most if not all universities worldwide offer introductory courses in number theory for math majors and in many cases as an elective course. 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. De nition Let a and b be integers. In either case, $m(m+1)(m+2)$ must be even. Prove that, for each natural number $n,$ $7^n-2^n$ is divisible by $5.$. 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 properties of divisibility, as they are known in Number Theory, states that: 1. Theorem. Example. 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. Show that the square of every of odd integer is of the form $8k+1.$, Exercise. The division of integers is a direct process. Now since both $(7^k-\cdot 2^k)$ and $7-2$ are divisible by 5, so is any linear combination of $(7^k- 2^k)$ and $7-2.$ Hence, $7^{k+1}-2^{k+1}$ is divisible by 5. Number theory, Arithmetic. Theorem. 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. Division Algorithm: Given integers a and b, with b > 0, there exist unique integers q and r satisfying a = qb+ r 0 r < b. The importance of the division algorithm is demonstrated through examples. Show that if $a$ is an integer, then $3$ divides $a^3-a.$, Exercise. Number Theory 1. 2. Discussion The division algorithm is probably one of the rst concepts you learned relative to the operation of division. Number Theory is one of the oldest and most beautiful branches of Mathematics. Recall we findthem by using Euclid’s algorithm to find \(r, s\) such that. Prove variant of the division algorithm. When a number $N$ is a factor of another number $M$, then $N$ is also a factor of any other multiple of $M$. Show $6$ divides the product of any three consecutive positive integers. First we prove existence. For integers a,b,c,d, the following hold: (a) aj0, 1ja, aja. You will see many examples here. Solution. 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. According to Wikipedia, “Number Theory is a branch of Pure Mathematics devoted primarily to the study of integers. About Dave and How He Can Help You. The division algorithm states that given an integer and a positive integer , there are unique integers and , with , for which . Thus, if we only wish to consider integers, we simply can not take any two integers and divide them. Show $3$ divides $a(a^2+2)$ for any natural number $a.$, Solution. 2. Definition. 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, $$ S=\{b i \mid i\in \mathbb{N} \text{ and } bi>a\}. Show that the product of two odd integers is odd and also show that the product of two integers is even if either or one of them is even. Some mathematicians prefer to call it the division theorem. 5 mod3 =5 3 b5 =3 c=2 5 mod 3 =5 ( 3 )b5 =( 3 )c= 1 5 mod3 = 5 3 b 5 =3 c=1 5 mod 3 = 5 ( 3 )b 5 =( 3 )c= 2 Be careful! We have $$ x a+y b=x(m c)+y(n c)= c(x m+ y n) $$ Since $x m+ y n \in \mathbb{Z}$ we see that $c|(x a+y b)$ as desired. If a number $N$ is divisible by $m$, then it is also divisible by the factors of $m$; 2. For any positive integer a and integer b, there exist unique integers q and r such that b = qa + r and 0 ≤ r < a, with r = 0 iﬀ a | b. 1. 0. Number Theory. Prove that the square of any integer is either of the form $3k$ or $3k+1.$, Exercise. In number theory, we study about integers, rational and irrational, prime numbers etc and some number system related concepts like Fermat theorem, Wilson’s theorem, Euclid’s algorithm etc. (Division Algorithm) If $a$ and $b$ are nonzero positive integers, then there are unique positive integers $q$ and $r$ such that $a=bq+r$ where $0\leq r < b.$. Use mathematical induction to show that $n^5-n$ is divisible by 5 for every positive integer $n.$, Exercise. A number of form 2 N has exactly N+1 divisors. Further Number Theory – Exam Worksheet & Theory Guides Example. The theorem does not tell us how to find the quotient and the remainder. Exercise. The same can not be said about the ratio of two integers. The notion of divisibility is motivated and defined. Learn number theory with free interactive flashcards. Lemma. $$ 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. Show that $f_n\mid f_m$ when $n$ and $m$ are positive integers with $n\mid m.$, Exercise. There are unique integers qand r, with 0 ≤r < d, such that a= dq+ r. All rights reserved. Edit. \[ 1 = r y + s n\] Then the solutions for \(z, k\) are given by. Also, if it is possible to divide a number $m$, then it is equally possible to divide its negative. Show that any integer of the form $6k+5$ is also of the form $3 j+2,$ but not conversely. 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 $a.$. Exercise. Lemma. Not to be confused with Euclid's division lemma, Euclid's theorem, or Euclidean algorithm. Let $m$ be an natural number. $$ 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. Proof. We call q the quotient, r the remainder, and k the divisor. 954−2 = 952. Proof. Proof. If $a|b,$ then $a^n|b^n$ for any natural number $n.$. Since c ∣ a and c ∣ b, then by definition there exists k1 and k2 such that a = k1c and b = k2c. Thus \(z\) has a unique solution modulo \(n\),and division makes sense for this case. If $a | b$ and $b |a,$ then $a= b.$. Use the PDF if you want to print it. Example. We will use mathematical induction. (Goldbach’s Conjecture) Is every even integer greater than 2 the sum of distinct primes? The next three examples illustrates this. Exercise. 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. Suppose $a|b$ and $b|a,$ then there exists integers $m$ and $n$ such that $b=m a$ and $a=n b.$ Notice that both $m$ and $n$ are positive since both $a$ and $b$ are. All 4 digit palindromic numbers are divisible by 11. We will use the Well-Ordering Axiom to prove the Division Algorithm. 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. (b) aj1 if and only if a = 1. A number other than1is said to be aprimeif its only divisors are1and itself. A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of Euclidean division. 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. This preview shows page 1 - 3 out of 5 pages. 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. (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. Certainly the sum, difference and product of any two integers is an integer. Division algorithms fall into two main categories: slow division and fast division. 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. Lemma. This characteristic changes drastically, however, as soon as division is introduced. The result will will be divisible by 7, 11 and 13, and dividing by all three will give your original three-digit number. Prove that the fourth power of any integer is either of the form $5k$ or $5k+1.$, 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 $7=2(2)+1(3).$. The Division Algorithm. 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 $y.$. Theorem 5.2.1The Division Algorithm Let a;b 2Z, with b 6= 0 . (d) If ajb and bjc, then ajc. Prove that the cube of any integer has one of the forms: $9k,$ $9k+1,$ $9k+8.$, Exercise. An integer other than There are integers $a,$ $b,$ and $c$ such that $a|bc,$ but $a\nmid b$ and $a\nmid c.$, Exercise. Examples. Therefore, $k+1\in P$ and so $P=\mathbb{N}$ by mathematical induction. Now we prove uniqueness. In this video, we present a proof of the division algorithm and some examples of it in practice. (Multiplicative Property of Divisibility) Let $a,$ $b,$ and $c$ be integers. Prove or disprove with a counterexample. The division algorithm describes what happens in long division. Any integer $n,$ except $0,$ has just a finite number of divisors. The Well-Ordering Axiom, which is used in the proof of the Division Algorithm, is then stated. 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. From the previous statement, it is clear that every integer must have at least two divisors, namely 1 and the number itself. The division algorithm, therefore, is more or less an approach that guarantees that the long division process is actually foolproof. If $a|m$ and $a|(ms+nt)$ for some integers $a\neq 0,$ $m,$ $s,$ $n,$ and $t,$ then $a|nt.$, Exercise. Suppose $a|b$ and $b|c,$ then there exists integers $m$ and $n$ such that $b=m a$ and $c=n b.$ Thus $$ c=n b=n(m a)=(n m )a.$$ Since $nm\in \mathbb{Z}$ we see that $a|c$ as desired. If $a | b$ and $b | c,$ then $a | c.$. That is, a = bq + r; 0 r < jbj. Using prime factorization to find the greatest common divisor of two numbers is quite inefficient. We call athe dividend, dthe divisor, qthe quotient, and r the remainder. Dave will teach you what you need to know, Applications of Congruence (in Number Theory), Diophantine Equations (of the Linear Kind), Euler’s Totient Function and Euler’s Theorem, Fibonacci Numbers and the Euler-Binet Formula, Greatest Common Divisors (and Their Importance), Mathematical Induction (Theory and Examples), Polynomial Congruences with Hensel’s Lifting Theorem, Prime Number Theorems (Infinitude of Primes), Quadratic Congruences and Quadratic Residues, Choose your video style (lightboard, screencast, or markerboard). http://www.michael-penn.net 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. Strictly speaking, it is not an algorithm. The proof of the Division Algorithm illustrates the technique of proving existence and uniqueness and relies upon the Well-Ordering Axiom. 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. Then we have $$ a=n b= n(m a) = (n m) a. We now state and prove the transitive and linear combination properties of divisibility. An algorithm describes a procedure for solving a problem. Prove that the cube of any integer has one of the forms: $7k,$ $7k+1,$ $7k-1.$, Exercise. The Division Algorithm. So the number of trees marked with multiples of 8 is Divisibility. (Division Algorithm) Given integers aand d, with d>0, there exists unique integers qand r, with 0 r0,$ then there exists unique integers $q$ and $r$ satisfying $a=bq+r,$ where $2b\leq r < 3b.$, Exercise. 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Long division for $ k=1 $ is divisible by 5 for every positive integer division! 0 and 6 as course notes for an undergraduate division algorithm number theory in number Theory simple lemmas. Solutions for \ ( n\ ), and k the divisor for $ k=1 is... For a more detailed explanation, please read the Theory Guides in section 2 below that given integer. The Well-Ordering Axiom topics that need to be ad-dressed in a course in number Theory states! D, the remainder unique integers and divide them main categories: slow and! Division is introduced “ number Theory flashcards on Quizlet and r so that a = +... Them in their personal and professional lives which is used in the book Elementary number Theory – Worksheet! Combinations and the number of divisors 3k $ or $ 3k+1. $, Exercise it any... Repeat a three-digit number twice, to form a six-digit number 5... While 2 and 3 are integers, we simply can not be said about the ratio $ $. J+2, $ the case for $ k=1 $ is also of the division algorithm is presented and.... That given an integer between 0 and 6 ) ajb and bjc, then acjbd $.