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Euclid: Elementa
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PROPOSITION 36. 
 
 
Given three numbers, to find the least number which they measure. 
 
 
Let A, B, C be the three given numbers;  thus it is required to find the least number which they measure. 
   
   
Let D, the least number measured by the two numbers A, B, be taken. [VII. 34]  Then C either measures, or does not measure, D.  First, let it measure it.  But A, B also measure D; therefore A, B, C measure D.  I say next that it is also the least that they measure. 
         
         
For, if not, A, B, C will measure some number which is less than D.  Let them measure E.  Since A, B, C measure E, therefore also A, B measure E.  Therefore the least number measured by A, B will also measure E. [VII. 35]  But D is the least number measured by A, B;  therefore D will measure E, the greater the less: which is impossible.  Therefore A, B, C will not measure any number which is less than D;  therefore D is the least that A, B, C measure. 
               
               
Again, let C not measure D,  and let E, the least number measured by C, D, be taken. [VII. 34]  Since A, B measure D, and D measures E, therefore also A, B measure E.  But C also measures E; therefore also A, B, C measure E.  I say next that it is also the least that they measure. 
         
         
For, if not, A, B, C will measure some number which is less than E.  Let them measure F.  Since A, B, C measure F, therefore also A, B measure F;  therefore the least number measured by A, B will also measure F. [VII. 35]  But D is the least number measured by A, B; therefore D measures F.  But C also measures F; therefore D, C measure F,  so that the least number measured by D, C will also measure F.  But E is the least number measured by C, D;  therefore E measures F, the greater the less: which is impossible.  Therefore A, B, C will not measure any number which is less than E.  Therefore E is the least that is measured by A, B, C.  Q. E. D. 
                       
                       
 
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