ClHtlOWS NUMERICAL PROPOSITION DEMONSTRATED, 



to contain a factor f, as Br: f B', which is not contained in 



a . b 

 A ', then I say, that — + — r-7 ~ n an integer, is impossi- 

 ble. 



^ J atB'-i-bA 



For —- + — ^p57 zz — r-^^^^ which cannot be an inle^ 



A — / B' A < B 



ger, because if it were, the numerator atlj' -\- b A would 

 be divisible by the denominator A t B', and consequently 

 by any one of the factors of that denominator, as t ; but 

 the numerator a< B' -}- i A is not divisible by ^ for since 

 the first term at B' is divisible by /, the second term b A 

 must also be divisible by /, if the whole quantity w^as so; 

 but b A is not divisible by t, (Euclid. 26, 7) because both b 

 and A are prime to t by the supposition ; since, then, atW 

 is divisible by t, but b A not divisible by it, therefore the 

 whale quantity «if B' + ^ A is not divisible by t, and conse- 



atB'±bA a , h 



quently 7—177 — or -r- + — ^ cahnot be equal to an 



A { Ju A — t si 



integer. ' Q. E. D. 



Cor. In the same manner it may be shewn, if there 

 be any number of fractions, each in their lowest terms, 



as -— > rg-> --^> &€,, and of which one of the denominators 

 ABC 



contain a factor that is not common to all the other deno- 

 minators, that neither the sum, nor difference of those 

 fractions, any how combined, can form an integer; that 



*^»"T'4Lir_7^_&0~""^" integer, is impossibleci 



Lemma 2, 



If any power of a number A, as A", be divisible by any Lemma i§i 

 other number r, once, and after that, neither by r, nor 

 by any factor of r, then will r itself be a complete n th 

 power. , . 



First if A be a prinme number, then A" can only be 

 divided by A, or some power of A, that is r must be some 

 power of A ; but if A" be divided by any power of A less 

 than A", it is evident that the quotient will still be divisible 



197 



