Proof That Two Numbers Divided by Their Greatest Common Divisor are Relatively Prime

Theory
If  
\[a, \; b\]
  are integers with greatest common divisor  
\[d\]
  then the greatest common divisor of  
\[\frac{a}{d}, \; \frac{b}{d}\]
  is 1.
Proof
If the greatest common divisor of  
\[a, \; b\]
  is d then we can find integers  
\[m, \; n\]
  such that  
\[am+bn=d\]
. Dividing by  
\[d\]
  gives  
\[\frac{a}{d}m+ \frac{b}{d}n=1\]
. Then by this theortem  
\[\frac{a}{d}, \; \frac{b}{d}\]
have greatest common divisor 1 and are relatively prime.

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