text: PROPOSITION 34.
  
  Given two numbers, to find the least number which they measure.
  
  Let A, B be the two given numbers; thus it is required to find the least number which they measure.
  
  Now A, B are either prime to one another or not.
  
  First, let A, B be prime to one another, and let A by multiplying B make C; therefore also B by multiplying A has made C. [VII. 16]
  
  Therefore A, B measure C
  
  I say next that it is also the least number they measure.
  
  For, if not, A, B will measure some number which is less than C.
  
  Let them measure D.
  
  Then, as many times as A measures D, so many units let there be in E, and, as many times as B measures D, so many units let there be in F; therefore A by multiplying E has made D, and B by multiplying F has made D; [VII. Def. 15] therefore the product of A, E is equal to the product of B, F.
  
  Therefore, as A is to B, so is F E. [VII. 19]
  
  But A, B are prime, primes are also least, [VII. 21] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less; [VII. 20] therefore B measures E, as consequent consequent.
  
  And, since A by multiplying B, E has made C, D, therefore, as B is to E, so is C to D. [VII. 17]
  
  But B measures E; therefore C also measures D, the greater the less: which is impossible.
  
  Therefore A, B do not measure any number less than C; therefore C is the least that is measured by A, B.
  
  Next, let A, B not be prime to one another, and let F, E, the least numbers of those which have the same ratio with A, B, be taken; [VII. 33] therefore the product of A, E is equal to the product of B, F. [VII. 19]
  
  And let A by multiplying E make C; therefore also B by multiplying F has made C; therefore A, B measure C.
  
  I say next that it is also the least number that they measure.
  
  For, if not, A, B will measure some number which is less than C.
  
  Let them measure D.
  
  And, as many times as A measures D, so many units let there be in G, and, as many times as B measures D, so many units let there be in H.
  
  Therefore A by multiplying G has made D, and B by multiplying H has made D.
  
  Therefore the product of A, G is equal to the product of B, H; therefore, as A is to B, so is H to G. [VII. 19]
  
  But, as A is to B, so is F to E.
  
  Therefore also, as F is to E, so is H to G.
  
  But F, E are least, and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less; [VII. 20] therefore E measures G.
  
  And, since A by multiplying E, G has made C, D, therefore, as E is to G, so is C to D. [VII. 17]
  
  But E measures G; therefore C also measures D, the greater the less: which is impossible.
  
  Therefore A, B will not measure any number which is less than C.
  
  Therefore C is the least that is measured by A, B. Q. E. D.