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Euclid: Elementa

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Euclid: Elementa
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PROPOSITION 29. 
 
 
Any prime number is prime to any number which it does not measure. 
 
 
Let A be a prime number, and let it not measure B;  I say that B, A are prime to one another. 
   
   
For, if B, A are not prime to one another, some number will measure them.  Let C measure them.  Since C measures B, and A does not measure B, therefore C is not the same with A.  Now, since C measures B, A, therefore it also measures A which is prime, though it is not the same with it: which is impossible.  Therefore no number will measure B, A.  Therefore A, B are prime to one another.  Q. E. D. 
             
             
 
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