Definition of relatively prime integers and Theorem on relatively prime integers.

 Definition: Two integers $a$ and $b$, not both of which are zero, are said to be relatively prime integers   if and only if $gcd(a, b) = 1$. 


Theorem: If $a$ and $b$ are any integers, not both them are zero, then $a$ and $b$ are relatively prime integers if and only if there exist integers $x$ and $y$ such that $a \cdot x +  b \cdot y = 1$.

Proof: Suppose that $a$ and $b$ are relatively prime integers, so that $gcd(a, b) = 1$. By Theorem, there exist integers $x$ and $y$ such that $a \cdot x + b \cdot y = 1$. conversely, suppose that there exist integers $x$ and $y$ such that $a \cdot x + b \cdot y = 1$ and that $d = gcd(a, b )$. Since $d = gcd(a,b)$, we have  $d ~|~ a$ and $d ~|~ b$. By part $7.$ of Theorem,  $d ~|~ (a  \cdot x + b \cdot y)$ for all integers $x, ~y$. So in particular  $d ~|~  (a \cdot x + b \cdot y) = 1$. Since $d$ is a positive integer and $d ~|~ 1$,  $d$ must be equal to $1$. Therefore $gcd(a, b ) = 1$ and hence $a$ and $b$ are relatively prime integers.

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