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Chapter II. THE INTEGERS

Chapter II. THE INTEGERS. 2.3 Divisibility 2.4 Prime and Greatest Common Divisor 2.5 Congruence of Integers 2.6 Congruence Classes. Divisibility. --- in the set of integers. Thm1: (The Well-Ordering Theorem; The Well-Ordering Principle).

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Chapter II. THE INTEGERS

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

  2. 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, mS such that m  x, xS.

  4. 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 multipleof 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 = 0x, 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. ab 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 -357=13·(-28)+7, that is, q = -28 and r = 7.

  8. Definition: • An integer d is calleda greatest common divisor（GCD）of integers a and b if 1. d is a positive integer. 2. daand 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).

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

  10. Ex2. Find (1776, 1492). • Sol:

  11. Ex3. Find (1400, -980). • Sol:

  12. 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.

  13. 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.

  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 primeif (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 abc, then ac. • Pf:

  17. Definition: • An integer p is called a primeif 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 pab,then pa or pb. • Pf:

  19. 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.

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