We call q the quotient, r the remainder, and k the divisor. Addition, subtraction, and multiplication follow naturally from their integer counterparts, but we have complications with division. Not to be confused with Euclid's division lemma, Euclid's theorem, or Euclidean algorithm. The algorithm that we present in this section is due to Euclid and has been known since ancient times. Definition. Thus, if we only wish to consider integers, we simply can not take any two integers and divide them. 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. These notes serve as course notes for an undergraduate course in number the-ory. Extend the Division Algorithm by allowing negative divisors. We call athe dividend, dthe divisor, qthe quotient, and r the remainder. Whence, $a^{k+1}|b^{k+1}$ as desired. Most if not all universities worldwide offer introductory courses in number theory for math majors and in many cases as an elective course. Therefore, $k+1\in P$ and so $P=\mathbb{N}$ by mathematical induction. $$ If $q_1=q_2$ then $r_1=r_2.$ Assume $q_1< q_2.$ Then $q_2=q_1+n$ for some natural number $n>0.$ This implies $$ 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. Any integer $n,$ except $0,$ has just a finite number of divisors. 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 $r$. Proof. The next lemma says that if an integer divides two other integers, then it divides any linear combination of these two integers. Now we prove uniqueness. We then give a few examples followed by several basic lemmas on divisibility. 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. Exercise. http://www.michael-penn.net 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 … 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. 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. It is probably easier to recognize this as division by the algebraic re-arrangement: 1. n/k = q + r/k (0 ≤ r/k< 1) That is, a = bq + r; 0 r < jbj. The proof of the Division Algorithm illustrates the technique of proving existence and uniqueness and relies upon the Well-Ordering Axiom. 954−2 = 952. There are other common ways of saying $a$ divides $b.$ Namely, $a|b$ is equivalent to all of the following: $a$ is a divisor of $b,$ $a$ divides $b,$ $b$ is divisible by $a,$ $b$ is a multiple of $a,$ $a$ is a factor of $b$. This preview shows page 1 - 3 out of 5 pages. His work helps others learn about subjects that can help them in their personal and professional lives. This is an incredible important and powerful statement. 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. The Division Algorithm. Prove that the cube of any integer has one of the forms: $9k,$ $9k+1,$ $9k+8.$, Exercise. The notes contain a useful introduction to important topics that need to be ad-dressed in a course in number theory. Show that if $a$ and $b$ are positive integers and $a|b,$ then $a\leq b.$, 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: Given integers a and b, with b > 0, there exist unique integers q and r satisfying a = qb+ r 0 r < b. Prove that the cube of any integer has one of the forms: $7k,$ $7k+1,$ $7k-1.$, Exercise. The same can not be said about the ratio of two integers. If we repeat a three-digit number twice, to form a six-digit number. In the book Elementary number theory by Jones a standard proof for division algorithm is provided. 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. The study of the integers is to a great extent the study of divisibility. His background is in mathematics and undergraduate teaching. Prove if $a|b,$ then $a^n|b^n$ for any positive integer $n.$, Exercise. According to Wikipedia, “Number Theory is a branch of Pure Mathematics devoted primarily to the study of integers. Since c ∣ a and c ∣ b, then by definition there exists k1 and k2 such that a = k1c and b = k2c. All rights reserved. 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. The theorem does not tell us how to find the quotient and the remainder. The total number of times b was subtracted from a is the quotient, and the number r is the remainder. The notion of divisibility is motivated and defined. Exercise. There are integers $a,$ $b,$ and $c$ such that $a|bc,$ but $a\nmid b$ and $a\nmid c.$, Exercise. The properties of divisibility, as they are known in Number Theory, states that: 1. 2. (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 $0\leq r < b.$. 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 $a | b$ and $b | c,$ then $a | c.$. The Well-Ordering Axiom, which is used in the proof of the Division Algorithm, is then stated. (Transitive Property of Divisibility) Let $a,$ $b,$ and $c$ be integers. Show that if $a$ is an integer, then $3$ divides $a^3-a.$, Exercise. Euclid’s Algorithm. We work through many examples and prove several simple divisibility lemmas –crucial for later theorems. A number other than1is said to be aprimeif its only divisors are1and itself. 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. Exercise. When a number $N$ is a factor of another number $M$, then $N$ is also a factor of any other multiple of $M$. [thm4] If a, b, c, m and n are integers, and if c ∣ a and c ∣ b, then c ∣ (ma + nb). 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. For integers a,b,c,d, the following hold: (a) aj0, 1ja, aja. Let's start off with the division algorithm. Further Number Theory – Exam Worksheet & Theory Guides Section 2.1 The Division Algorithm Subsection 2.1.1 Statement and examples. The result will will be divisible by 7, 11 and 13, and dividing by all three will give your original three-digit number. Add some text here. 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 $a.$. Number theory, Arithmetic. $$ Thus, $n m=1$ and so in particular $n= 1.$ Whence, $a= b$ as desired. If $a|m$ and $a|(ms+nt)$ for some integers $a\neq 0,$ $m,$ $s,$ $n,$ and $t,$ then $a|nt.$, Exercise. If $a | b$ and $b |a,$ then $a= b.$. Lemma. In addition to showing the divisibility relationship between any two non zero integers, it is worth noting that such relationships are characterized by certain properties. The natural number $m(m+1)(m+2)$ is also divisible by 3, since one of $m,$ $m+1,$ or $m+2$ is of the form $3k.$ Since $m(m+1)(m+2)$ is even and is divisible by 3, it must be divisible by 6. Prove that $7^n-1$ is divisible by $6$ for $n\geq 1.$, Exercise. Similarly, $q_2< q_1$ cannot happen either, and thus $q_1=q_2$ as desired. If a number $N$ is a factor of two number $s$ and $t$, then it is also a factor of the sum of and the difference between $s$ and $t$; and 4. Discussion The division algorithm is probably one of the rst concepts you learned relative to the operation of division. Number Theory is one of the oldest and most beautiful branches of Mathematics. Show that the square of every of odd integer is of the form $8k+1.$, Exercise. First we prove existence. Zero is divisible by any number except itself. If $a$ and $b$ are integers with $a\neq 0,$ we say that $a$ divides $b,$ written $a | b,$ if there exists an integer $c$ such that $b=a c.$, Here are some examples of divisibility$3|6$ since $6=2(3)$ and $2\in \mathbb{Z}$$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}$$-9|909$ since $909=-101(-9)$ and $-101\in \mathbb{Z}$. (e) ajb and bja if and only if a = b. Arithmetic - Arithmetic - Theory of divisors: At this point an interesting development occurs, for, so long as only additions and multiplications are performed with integers, the resulting numbers are invariably themselves integers—that is, numbers of the same kind as their antecedents. If $c|a$ and $c|b,$ then $c|(x a+y b)$ for any positive integers $x$ and $y.$. Copyright © 2021 Dave4Math LLC. The importance of the division algorithm is demonstrated through examples. Before we state and prove the Division Algorithm, let’s recall the Well-Ordering Axiom, namely: Every nonempty set of positive integers contains a least element. [June 28, 2019] These notes were revised in Spring, 2019. All 4 digit palindromic numbers are divisible by 11. We will use the Well-Ordering Axiom to prove the Division Algorithm. Just for context here is Theorem 1.1: If $a$ and $b$ are integers with $b > 0$, then there is a unique pair of integers $q$ and $r$ such that $$a=qb+r$$ and $$0\le r < … For example, when a number is divided by 7, the remainder after division will be an integer between 0 and 6. $$ 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. Solution. You will see many examples here. (Multiplicative Property of Divisibility) Let $a,$ $b,$ and $c$ be integers. 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). Show that $f_n\mid f_m$ when $n$ and $m$ are positive integers with $n\mid m.$, Exercise. \[ z = x r + t n , k = z s - t y \] for all integers \(t\). 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. Proof. Use mathematical induction to show that $n^5-n$ is divisible by 5 for every positive integer $n.$, Exercise. This characteristic changes drastically, however, as soon as division is introduced. 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. Theorem 5.2.1The Division Algorithm Let a;b 2Z, with b 6= 0 . Using prime factorization to find the greatest common divisor of two numbers is quite inefficient. (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. Divisibility. Solution. 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