CURIOUS PROPERTY OF SINGLE REPETEND8. 125 



PROPOSITION. Propositi^. 



EVF.RY odd number, except 5 audits multiples, is a di- 

 visor of a repetend of any of the nine digits; and the num- 

 ber of digits necessary to form the dividend will never ex- 

 ceed the number expressed by the divisor. 



Demonstration, 



First it is evident, that, if we can prove the truth of this Demonstra- 

 proposition for a repetend of units, it must necessarily be 

 true also for a repetend of any other digit, since such repe- 

 tend would be a multiple of a repetend of units. 



Again if the former part of the proposition be true, the 

 truth of the latter part also easily follows ; for if 111111, &c. 

 be divisible by any number D, no remainder can recur, till 

 after the remainder has occurred ; since, if any remainder 

 recurred before the division terminated, the operation would 

 proceed with precisely the fame figures as when that remain- 

 der first occurred ; and thus this remainder would recur 

 again, and so on ad infinitum ; and hence the division would 

 never terminate, or 111111 &c. would not be divisible by 

 I), contrary to hypothesis. Now since all the possible re- 

 mainders, which can occur in dividing 111111 &c. by D, 

 will be between D and inclusive, therefore there can be 

 but D different remainders; and since, in the operation of 

 division, each figure in the dividend will give one remainder, 

 therefore D figures in the dividend will give D remainders. 

 Hence in dividing 111111 &c. to D places of digits by D, all 

 the different remainders, which can take place, will occur; 

 and therefore, if the dividend were to consist of more digits 

 than are denoted by the divisor, some one or more of the re- 

 mainders would recur; and hence if the division did not 

 terminate precious to this recurrence, it would never termi- 

 nate, but would goon ad infinitum. Consequently, if ] 1 1111 

 &c. be ever divisible by D, it must be so when or before the 

 dividend consists of D digits. We say before, because, 

 though the remainder might not occur precisely at the end 

 •f D digits, yet, from what has been shevvn above, if that 



remainder 



