1 13 Interesting Pi^operties of Numbers. 



Art. XL — Development of some interesting Properties of Num- 

 bers ; by George R. Perkins. 



If we multiply a unit by any number N, and divide the result 

 by a number P, then multiply the remainder by N, and again 

 divide by P ; and thus continue to multiply?" the remainder by 

 N, and to divide by P : we shall obtain a succession of quotients 

 and remainders which we will represent by ^i, g,, q^, • ■ • qx 

 and r,, r^, Tg, . . . ?\. 



From the above law of operation, we readily deduce the fol- 

 lowing equations : 



•j^_p -.^ From the first of these equations we 



T^~ __p / , can find r^, which substituted in the se- 



^ ' ~-p T ^ [ f-i\ cond will make known r,, which in turn 





^ ^ substituted in the third will give ?% ; and 

 thus we may continue until we have ob- 

 tained the following equations : 



r,-N-Pg, 



r,=N^-P[%,+^,] 



r3=N3-P[N^g,4-%3+^3] 



k2) 



r.=N^-P[N^-'^,+N^--^,+ . . . +N^_,+^,] 

 Since r^ in the general equation of (2), is less than P, it fol- 

 lows that if we divide that equation by P, the remainder on the 

 left hand side of the equation will be r^ ; and consequently we 

 must have the same remainder on the right. Now, since the 

 term within the brackets is multiplied by P, it can leave no re- 

 mainder when divided by P: hence we conclude. 



That N"" divided by P will give r^ for remainder. 

 If in the general equation (2), we substitute M for the expres- 

 sion within the brackets, we shall obtain r,,=N'' — PM (3), this 

 being true for all values of x, we shall also have r^,=N''' -PM' 

 (4). Multiplying (3) and (4) together, we get r, Xr,/=N"+"'-P 

 [MN^'+M'N^-PMM^] (5). Hence we conclude. 



That r ^ y.r^i divided by P, will give r^^^ifor remddnder. 

 From the general equation of ( 1 ) we discover, 

 That (N— RJQ',^. divided by N, will give the same remmnder as 

 r ^ divided by N, ivhere R is the remainder of P divided by N. 



It is evident that the process will terminate whenever we ob- 

 tain /\,=0 ; but when this is not the case, the quotients and re- 



