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Euclid: Elementa

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Euclid: Elementa
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PROPOSITION 35. 
 
 
If two numbers measure any number, the least number measured by them will also measure the same. 
 
 
For let the two numbers A, B measure any number CD, and let E be the least that they measure;  I say that E also measures CD. 
   
   
For, if E does not measure CD, let E, measuring DF, leave CF less than itself.  Now, since A, B measure E, and E measures DF, therefore A, B will also measure DF.  But they also measure the whole CD;  therefore they will also measure the remainder CF which is less than E: which is impossible.  Therefore E cannot fail to measure CD;  therefore it measures it.  Q. E. D. 
             
             
 
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