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

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