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. --- 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, 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 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 = 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 =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 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).

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, = 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 abc, then ac.
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 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.