Chapter II. THE INTEGERS 2.3 Divisibility 2.4 Prime and Greatest Common Divisor 2.5 Congruence of Integers 2.6 Congruence Classes

--- in the set of integers Divisibility --- in the set of integers

Thm1: (The Well-Ordering Theorem; The Well-Ordering Principle) Every nonempty set S of positive integers contains a least element. That is, mS such that m  x, xS.

Definition: Let a, b  Z. We say a divides b if c  Z such that b = ac, denoted by ab. If ab, then we say that b is a multiple of a and a is a factor (or divisor) of b.

Theorem 2: The only divisors of 1 are 1 and -1. Note: 0 is a multiple of every integer in Z. Since 0 = 0x, x Z.

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. ab if and only if r = 0

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 -357=13·(-28)+7, that is, q = -28 and r = 7.

Definition: An integer d is called a greatest common divisor （GCD）of integers a and b if 1. d is a positive integer. 2. da and db. 3. If ca and cb, then cd. Note: We usually denote the GCD d of a and b by d = (a, b).

Use the Euclidean algorithm to find (a, b).

Ex2. Find (1776, 1492). Sol:

Ex3. Find (1400, -980). Sol:

Theorem4: Let a, bZ, 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.

Ex2’. Find integers m and n such that 4 = 1776m+ 1492n. By Ex2. We have1776 = 1492·1+284, 1492 = 284·5+72, 284 = 72·3+68, 72 = 68·1+4.

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.

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.

Theorem 5. Let a, b and c be integers. If a and b are relatively prime and abc, then ac. Pf:

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.

Thm6: (Euclid’s Lemma) Let a and b be integers. If p is a prime and pab, then pa or pb. Pf:

Cor7: Let a1, a2, …, an be integers and p is a prime. If pa1a2···an , then paj for some j = 1, 2, …, n.