Brief review of notions from elementary number theory concerning the set

Z = {..., -2, -1, 0, 1, 2...} of integers and
N = {0, 1, 2 , ...} of natural numbers.
Z_n= {0, 1, 2… n-1}
Z_n⁺= {1, 2… n-1.}

Common divisors and greatest common divisors (GCD):

Let a, b ∈ Z

d ∈ Z Λ d | a  Λ  d | b ⇒ d |  ax  +  by [ x, y  ∈ Z]

Let d = gcd (a, b)

The following are elementary properties of the gcd function:

gcd( a , b ) = gcd( b , a )

gcd( a , b ) = gcd(- a , b )

gcd( a , b ) = gcd(| a |, | b |)

gcd( a , 0) = | a |

gcd( a , ka ) = | a | for any k ∈ Z .

Theorem 1
If a and b are any integers then gcd(a,b) is the smallest positive element of the set {ax + by : x, y ∈ Z }

Proof:

Let s be the smallest positive element of the set:{ ax + by : x , y ∈Z}

Let q = ⌊ a / s ⌋ and s = ax + by

a mod s = a – qs = a - q ( ax + by ) = a (1 - qx ) + b (- qy )

a mod s < s and a mod s is a linear combination of a and b . Thus a mod s = 0 ⇒ s | a

Using analogous reasoning we can show s | b. Thus s ≤ gcd (a,b ).

Let d = gcd (a,b) ⇒ d | a and d | b. Thus d | s and s >0 ⇒ d ≤ s. We have shown before d ≥ s and thus we have established that d=s.