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2014 년 봄학기 강원대학교 컴퓨터과학전공 문양세

이산수학 (Discrete Mathematics)  정수와 나눗셈 (The Integers and Division). 2014 년 봄학기 강원대학교 컴퓨터과학전공 문양세. Introduction. The Integers and Division. Of course you already know what the integers are, and what division is… ( 정수와 나누기 … 다 아는거 ~)

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2014 년 봄학기 강원대학교 컴퓨터과학전공 문양세

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  1. 이산수학(Discrete Mathematics) 정수와 나눗셈(The Integers and Division) 2014년 봄학기 강원대학교 컴퓨터과학전공 문양세

  2. Introduction The Integers and Division • Of course you already know what the integers are, and what division is… (정수와 나누기… 다 아는거~) • But: There are some specific notations, terminology, and theorems associated with these concepts which you maynot know. (몇 가지 주요 표기, 용어, 정리 등을 배웁시다.) • These form the basics of number theory. (정수론의 기초..) • Vital in many important algorithms today (hash functions, cryptography, digital signatures).

  3. Divides, Factor, and Multiple The Integers and Division • Let a,bZ with a0. • a|b  “adividesb” : “cZ:b=ac”“There is an integer c such that c times a equals b.”(b가 a로 나누어짐(a로 b를 나눌 수 있음)을 의미하며, 이때 몫이 c이다.) • Example: 312  True, but 37  False. • If a divides b, then we say a is a factor or a divisor of b, and b is a multiple of a. • “b is even” :≡ 2b. Is 0 even? Is −4? 인수 배수

  4. Some Facts of Division The Integers and Division • a,b,c  Z: 1. a|0 (2|0, 3|0, …) 2. (a|b a|c)  a | (b + c) (2|4  2|6  2|10) 3. a|b  a|bc (2|4  2|4∙3) 4. (a|b b|c)  a|c (2|4  4|8  2|8) • Proof of (2): • a|b means there is an s such that b=as, • and a|c means that there is a t such that c=at, • so b+c = as+at = a(s+t), so a|(b+c) also.□

  5. More Detailed Version of Proof The Integers and Division Show a,b,c  Z: (a|b a|c)  a | (b + c). Let a, b, c be any integers such that a|band a|c, and show that a | (b + c). By definition of |, we know s: b=as, and t: c=at.Let s, t, be such integers. Then b+c = as + at = a(s+t)so u: b+c=au, namely u=s+t.Thus a|(b+c).

  6. Prime Numbers (소수) The Integers and Division An integer p>1 is prime iff it is not the product of any two integers greater than 1:p>1  a,bN:a>1, b>1, ab=p. The only positive factors of a prime p are 1 and p itself. Some primes: 2,3,5,7,11,13...(양의 인수(divisor)가 1과 자기 자신뿐이면 소수(prime)라 한다.) Non-prime integers greater than 1 are called composite, because they can be composed by multiplying two integers greater than 1. 합성수

  7. Fundamental Theorem of Arithmetic(산술의 기본 정리) The Integers and Division • Every positive integer has a unique representation as the product of a non-decreasing series of zero or more primes.(모든 양의 정수는 오름차순으로 정렬된 소수들의 곱으로 유일하게 표현된다.결국 Prime Factorization(소인수분해)를 이야기한다고 볼 수 있다.) • 1 = (product of empty series) = 1 • 2 = 2 (product of series with one element 2) • 4 = 2·2 (product of series 2,2) • 2000 = 2·2·2·2·5·5·5 = 24·53; 2001 = 3·23·29;2002 = 2·7·11·13; 2003 = 2003

  8. 소수의 개수? The Integers and Division • “무한히 많은 소수가 존재함”을 증명하라. 1. 유한 개의 소수 p1, p2, …, pn이 있다고 가정하자. 2. Q = p1·p2·…·pn + 1이라 하자. 3. “산술의 기본 정리”에 의하여 Q는 소수들의 곱으로 표현된다. 4. 따라서, pj | Q(1  j  n)를 만족하는 소수 pj가 존재한다. 5. 그러면, 다음과 같은 전개가 이루어진다. 5.1 (pj | p1·p2·…·pn)  (pj |-(p1·p2·…·pn)) [a|b  a|bc] 5.2 (pj | Q) = (pj | (p1·p2·…·pn+ 1)) [by step 4] 5.3 (pj | 1)? [(a|b a|c)  a | (b + c)] 6.pj는1을 나눌 수 없으므로, 결국 가정(유한 개의 소수가 있다)이 잘못된 것이다.즉, 소수의 개수는 무한하다. □

  9. The Division “Algorithm” The Integers and Division • Really just a theorem, not an algorithm… • The name is used here for historical reasons. • For any integer dividenda and divisord≠0, there is a unique integer quotientq and remainder rNsuch that a = dq + r and 0  r < |d|. • dividend = “피제수”, divisor = “제수” • quotient = “몫”, remainder = “나머지” • a,dZ, d>0:!q,rZ: 0r<|d|, a=dq+r.(! means “unique”) • We can find q and r by: q=ad, r=aqd.(e.g., if a = 14 and d = 3, then q = 143 = 4 and r = 14 - 3·4 = 2.

  10. Greatest Common Divisor (최대공약수) The Integers and Division The greatest common divisor gcd(a,b) of integers a,b (not both 0) is the largest (most positive) integer d that is a divisor both of a and of b.(최대공약수는 두 수의 공약수 중에서 가장 큰 공약수이다.) d = gcd(a,b) = max(d: d|a  d|b) Example: gcd(24,36)=?Positive common divisors: 1,2,3,4,6,12Greatest is 12.

  11. GCD shortcut The Integers and Division • If the prime factorizations are written as and ,then the GCD is given by: • Example: • a=84=2·2·3·7 = 22·31·71 • b=96=2·2·2·2·2·3 = 25·31·70 • gcd(84,96) = 22·31·70 = 2·2·3 = 12.

  12. Relatively Primality (서로 소, 상대 소수) The Integers and Division • Integers a and b are called relatively prime or coprime iff their gcd = 1.(두 수의 최대공약수가 1인 경우, 두 수는 서로 소(상대 소수)라 한다.) • Example: Neither 21 and 10 are prime, but they are coprime. 21=3·7 and 10=2·5, so they have no common factors > 1, so their gcd = 1. • A set of integers {a1,a2,…} is (pairwise) relatively prime if all pairs ai, aj, ij, are relatively prime.(주어진 정수들의 모든 쌍이 서로 소이면, 해당 정수들은 서로 소라 한다.) • E.g., 10, 17, and 21 are relatively primes.

  13. Least Common Multiple (최소공배수) The Integers and Division • lcm(a,b) of positive integers a, b, is the smallest positive integer that is a multiple both of a and of b.m = lcm(a,b) = min(m: a|m b|m)(최소공배수는 두 수의 공배수 중에서 가장 작은 공배수이다.) • E.g. lcm(6,10)=30 • If the prime factorizations are written as and , then the LCM is given by

  14. A Useful Rule The Integers and Division If a and b are positive integers,then ab = gcd(a,b)·lcm(a,b) Prove it by yourself!

  15. The mod Operator The Integers and Division • An integer “division remainder” operator. • Let a,dZ with d>1, then • amodd denotes the remainder r; • i.e. the remainder when a is divided by d. • r = amodd • We can compute (amodd) by: a  d·a/d. • In C programming language, “%” = mod. q

  16. Modular Congruence (모듈로 합동?) The Integers and Division • Let Z+={nZ| n>0}, the positive integers. • Let a,bZ, mZ+. • Then ais congruent tob modulo m, written “ab(mod m)”,iffm|(ab). • Also equivalent to: (ab) mod m = 0. • m으로 나누었을 때, a와 b의 나머지가 동일(합동)하다. • m으로 나누어 나머지가 동일한 수의 차를 m으로 나누면 나머지가 0이다. 두 수의 차를 m으로 나눌 수 있으면, 두 수를 m으로 나눈 나머지는 동일하다. • Example: 175(mod 6) 6|(17–5), (17–5)%6 = 0

  17. Spiral Visualization of mod The Integers and Division Example shown:modulo-5arithmetic ≡ 0 (mod 5) 20 15 10 ≡ 1 (mod 5) 5 21 19 ≡ 4 (mod 5) 14 16 9 0 11 4 6 1 3 2 8 7 13 12 18 17 ≡ 2 (mod 5) 22 ≡ 3 (mod 5) 5로 나누어 나머지가 3인 수

  18. Useful Congruence Theorems The Integers and Division 나머지 • Let a,bZ, mZ+. Then:ab (mod m)  kZa=b+km. • Let a,b,c,dZ, mZ+. Then if ab (mod m) and cd (mod m), then: (a와 b가 m 모듈로 합동이고, c와 d가 m 모듈로 합동이면,) • a+c b+d (mod m), and • ac  bd (mod m) 몫

  19. Applications of Congruence (합동의 응용) The Integers and Division • The mod operator is widely used in hash functions. • h(key) = key mod m • Linear congruential methods is used to generate pseudo random numbers. • x[n+1] = (a·x[n] + c) mod m (생략) • Also, in cryptography, encryption, …

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