Quiz 1 (1.3)

Question 1

Natural numbers

Answer 1

Is the set where

N={0,1,2,3,…}

Question 2

Integers

Answer 2

The set of all integers positive, negative, and 0 is denoted by
Z={+-n; n€N}

Question 3

Least Integer Axiom (Well-Ordering Axiom)

Answer 3

States that every nonempty collection C of natural numbers contains a smallest element; that is, there’s a number c°€C with c° <= c for all c € C

Question 4

Divides

Answer 4

If a and b are integers, then a divided b denoted by a|b, if there’s an int c with b=ca. We can also say that a is a divisor (or a factor) of, and that b is a multiple of a

Question 5

Prime

Answer 5

An int a is a prime if a >= and its only divisors are +- 1 and +-2: if a>= 2 has other divisors, then it’s called composite

Question 6

Composite

Answer 6

If a >= 2 has other divisors (other than +-1 and +-a

Question 7

Division algorithm

Answer 7

If a and b are positive integers, then there are unique (i.e. exactly one) integers q (the quotient) and r (the remainder) with b=qa+r and 0<=r<a></a>

Question 8

Common divisor

Answer 8

The common divisor of integers a and b is an integer c with c/a and c/b

Question 9

Greatest common divisor

Answer 9

The greatest common divisor of a and b, denoted by gcd(a,b) or more briefly, by (a,b) (or more briefly, by (a,b)), is defined by

Gcd(a,b)={
0 if a=0=b
The largest common divisor of a and b o.w.
}

Question 10

Linear combination

Answer 10

A linear combination of integers a and b is an integer of the form sa+tb where s,t€Z (the numbers s,t are allowed to be negative ).

Question 11

Corollary 1.20

Answer 11

Let a and b be integers a nonnegative common divisor d is their gcd iff c|d for every common divisor c of a and b

Question 12

Euclids lemma Thm 1.21

Answer 12

If p is a prime and p|ab for integers a,b, then p|b. Conversely, if m>=2 is an int s.t. m|ab always implies m|a or m|b, then m is a prime

Question 13

Relatively prime

Answer 13

Call integers a and b relatively prime if their gcd is 1

Question 14

Proposition 1.23

Answer 14

Let a and b be ints.
(i) gcd(a,b)=1 (that is, a and b are relatively prime) iff 1 is a linear combination of a and b

(ii) if d=gcd(a,b)=/ 0, then the ints a/d and b/d are relatively prime

Question 15

Primitive

Answer 15

A Pythagorean triple (a,b,c) is primitive if a,b,c have no common divisor d>=2; that is, there is no int d>=2 which divides each of a,b, and c.