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Euclid: Elementa

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Euclid: Elementa
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PROPOSITION 26. 
 
 
If two numbers be prime to two numbers, both to each, their products also will be prime to one another. 
 
 
For let the two numbers A, B be prime to the two numbers C, D; both to each, and let A by multiplying B make E, and let C by multiplying D make F;  I say that E, F are prime to one another. 
   
   
For, since each of the numbers A, B is prime to C,  therefore the product of A, B will also be prime to C. [VII. 24]  But the product of A, B is E;  therefore E, C are prime to one another.  For the same reason E, D are also prime to one another.  Therefore each of the numbers C, D is prime to E.  Therefore the product of C, D will also be prime to E. [VII. 24]  But the product of C, D is F.  Therefore E, F are prime to one another.  Q. E. D. 
                   
                   
 
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