Presentation on theme: "Chapter II. THE INTEGERS"— Presentation transcript:
1 Chapter II. THE INTEGERS 2.3 Divisibility2.4 Prime and Greatest Common Divisor2.5 Congruence of Integers2.6 Congruence Classes
2 --- in the set of integers Divisibility--- in the set of integers
3 Thm1: (The Well-Ordering Theorem; The Well-Ordering Principle) Every nonempty set S of positive integers contains a least element.That is, mS such that m x, xS.
4 Definition:Let a, b Z. We say a divides b if c Z such that b = ac, denoted by ab.If ab, then we say that b is a multiple of a and a is a factor (or divisor) of b.
5 Theorem 2: The only divisors of 1 are 1 and -1. Note: 0 is a multiple of every integer in Z.Since 0 = 0x, x Z.
6 Thm3: (The Divisor Algorithm; The Euclidean Algorithm) Let a, b Z with b>0. Then ! q and r in Z such that a = bq + r where 0 r < b.Note:1. q = quotient (could be any integer);r = remainder ( 0)2. ab if and only if r = 0
7 Ex1.Let a =357 and b =13. Then 357=13·27+6,that is, q = 27 and r = 6.If a = -357 and b =13, then =13·(-28)+7,that is, q = -28 and r = 7.
8 Definition:An integer d is called a greatest common divisor （GCD）of integers a and b if1. d is a positive integer.2. da and db.3. If ca and cb, then cd.Note: We usually denote the GCD d of a and b by d = (a, b).
12 Theorem4:Let a, bZ, not both 0. Then there is a GCD d in Z of a and b.Moreover, d = am + bn for integers m, n.The positive integer d is the smallest positive integer that can be written in this form.
13 Ex2’. Find integers m and n such that 4 = 1776m+ 1492n. By Ex2. We have1776 = 1492·1+284, 1492 = 284·5+72, = 72·3+68, 72 = 68·1+4.
14 Ex3’. Find m, n such that 140 = 1400m + (-980)n. From Ex3, we have 1400 = (-980)·(-1)+420-980 = 420·(-3)+280, 420 = 280·1+140.
15 Definition: Two integers a and b are relatively prime if (a, b) = 1. For instance:(2, 5) = 1,thus 2 and 5 are relatively prime.(-2, -5) =1,thus -2 and -5 are relatively prime.
16 Theorem 5.Let a, b and c be integers. If a and b are relatively prime and abc, then ac.Pf:
17 Definition:An integer p is called a prime if p > 1 and the only divisors of p are 1 and p.For instance:2 and 5 are both primes.But –2 and 6 are not primes.Note: 1 is not a prime.
18 Thm6: (Euclid’s Lemma)Let a and b be integers. If p is a prime and pab, then pa or pb.Pf:
19 Cor7:Let a1, a2, …, an be integers and p is a prime. If pa1a2···an , then paj for some j = 1, 2, …, n.