Section 2 - Integers

Question 1

Definition of Divisibility

Answer 1

If a,b∈Z, then b is said to divide a if there is c∈Z such that a = bc (we write b|a)

Question 2

Division with Remainder Proposition

Answer 2

Let a,b∈Z with b != 0. Then there are unique q,r∈Z with 0 <= r < |b| such that a = qb + r (where q is the quotient and r is the remainder)

Question 3

Greatest Common Divisor

Answer 3

A positive common divisor of a and b that is divisible by all common divisors

Question 4

GCD Theorem

Answer 4

Let a,b∈Z, not both 0. The following hold:
(i) A GCD of a and b exists, and it is unique
(ii) There are integers m,n∈Z such that gcd(a, b) = ma + nb

Question 5

Prime Number

Answer 5

An integer > 1 whose only positive divisors are 1 and itself

Question 6

Unique Factorization Theorem

Answer 6

Let p be a prime number. If a,b∈Z and p|ab, then p|a or p|b

Question 7

Fundamental Theorem of Arithmetic

Answer 7

Every positive integer can be factorized into a product of primes, and the factorization is unique up to the order of the prime numbers (e.g. 2x3=6 and 3x2=6 are unique factorizations, but here they are considered 1 factorization)

Question 8

Coprime

Answer 8

Two integers a and b are said to be coprime if gcd(a,b) = 1