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**Chapter II. THE INTEGERS**

2.3 Divisibility 2.4 Prime and Greatest Common Divisor 2.5 Congruence of Integers 2.6 Congruence Classes

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**--- in the set of integers**

Divisibility --- in the set of integers

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**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.

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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.

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**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.

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**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

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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.

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Definition: An integer d is called a greatest common divisor （GCD）of integers a and b if 1. 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).

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

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Ex2. Find (1776, 1492). Sol:

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

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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.

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**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.

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**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.

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**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.

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

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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.

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

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Cor7: Let a1, a2, …, an be integers and p is a prime. If pa1a2···an , then paj for some j = 1, 2, …, n.

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