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Euclid: Elementa
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PROPOSITION 23. 
 
 
If two number be prime to one another, the number which measures the one of them will be prime to the remaining number. 
 
 
Let A, B be two numbers prime to one another, and let any number C measure A;  I say that C, B are also prime to one another. 
   
   
For, if C, B are not prime to one another, some number will measure C, B.  Let a number measure them, and let it be D.  Since D measures C, and C measures A, therefore D also measures A.  But it also measures B;   therefore D measures A, B which are prime to one another: which is impossible. [VII. Def. 12]  Therefore no number will measure the numbers C, B.  Therefore C, B are prime to one another.  Q. E. D. 
               
               
 
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