won2 - WON Series in Discrete Mathematics and Modern Algebra Volume 2 NUMBER THEORY Amin Witno Preface Written at Philadelphia University Jordan for

won2 - WON Series in Discrete Mathematics and Modern...

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WON Series in Discrete Mathematics and Modern Algebra Volume 2 NUMBER THEORY Amin Witno Preface Written at Philadelphia University, Jordan for Math 313, these notes 1 were used first time in the Fall 2005 semester. They have since been revised 2 and shall be revised again as often as the author teaches the course. Outline notes are more like a revision. No student is expected to fully benefit from these notes unless they have regularly attended the lectures. 1 Divisibility The natural numbers 1 , 2 , 3 , . . . together with their negatives and zero make up the set of integers. Number Theory is the study of integers. Every number represented throughout these notes will be understood an integer unless otherwise stated. Definition. The number d divides m , or m is divisible by d , if the operation m ÷ d yields an integer. This relation may be written d | m , or d | m if it is not true. When d | m , the number d is also called a divisor of m , and m a multiple of d . For example 3 | 18 and 5 | 18. We may also state that even numbers are multiples of 2. 1.1 Proposition (Properties of divisibility) 1. The number 1 divides all integers. 2. d | 0 and d | d for any integer d = 0. 3. If d | m and m | n then d | n . 4. If d | m and d | n then d | ( am + bn ) for any integers a and b . Proof. The first two statements follow immediately from the definition of divisibility. For (3) simply observe that if m/d and n/m are integers then so is n/d = n/m × m/d . Similarly for (4), the number ( am + bn ) /d = a ( m/d ) + b ( n/d ) is an integer when d | m and d | n . Definition. For every real number x , the notation x denotes the greatest integer x . For example 3 . 14 = 3 and 2 = 2. Now with n > 0, define the residue of m mod n by m % n = m - m/n n . Here the symbol (%) is read “mod”. For example 18 % 5 = 3 1 Copyrighted under a Creative Commons License c 2005–2007 Amin Witno 2 Last Revision: 24–12–2006 www . w i t n o . com
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WON 2 – Number Theory 2 and 18 % 3 = 0. Note that m % n is really the remainder upon dividing m by n and it lies in the range 0 m % n n - 1. In particular m % n = 0 if and only if n | m . Exercise. Find these residues. 1. 369 % 5 2. 24 % 8 3. 123456789 % 10 4. 7 % 11 1.2 Proposition One in every n consecutive integers is divisible by n . Proof. Let m be the first integer and let k = m % n . If k = 0 then n | m . Otherwise 1 k n - 1 and our consecutive integers can be written m = m/n n + k, m/n n + ( k + 1) , m/n n + ( k + 2) , . . . , m/n n + ( k + n - 1) with k + n - 1 n . Then one of these numbers is m/n n + n , a multiple of n . Definition. The greatest common divisor of two integers m and n is the largest integer which divides both. This number is denoted by gcd( m, n ). For example gcd(18 , 24) = 6 because 6 is the largest integer with the property 6 | 18 and 6 | 24. Exercise. Find gcd(36 , - 48) , gcd(24 , 0) , gcd(1 , 99) , gcd(100 , 123). 1.3 The Euclidean Algorithm gcd( m, n ) = gcd( n, m % n ) Proof. It suffices to show that the two pairs { m, n } and { n, m % n } have identical sets of common divisors. This is achieved entirely using Proposition 1.1.4 upon observing that, from its definition, m % n is a linear combination of m and n , and so is m of n and m % n .
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