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“2 Uwe ee of Numbers. 
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a XL.—Development of some interesting hee ae of Num 
bers ; by Gzorce R. Perkins 
Ir we multiply a unit by any number N, and divide the result 
by anumber 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 4,, 2; 19 ow 3 
and r,, 7, : 
From the abot 1a of operation, We véailily doauas the fol- 
lowing equations : %. : 
: ‘From the first of these equations we 
can find r,, which substituted in the se- 
: ° cond will make known r,, which in tum 
= Sere (1) substituted in the third will give r, ; and 
N = =Pg = thus we may continue until we have ob- 
. ° tained the following equations: 
r,=N—- Pq, : : 
r,=N?- PING, +q,] : 
e, — : Fie q +N9, +4.] — >(2) 
,=Ne _P[N-- Mp ANG. : ENG EEA 
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 1 no fe- 
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 =Nte-P 
[MN*'+M’N*—PMM’] (5). Hence we conclude, 
That r, Xr: divided by P, will giver, ., for remaiadlels 
From the general equation of (1) we discover, 
That (N—R)q. divided by N, will give the same remainder 
r, divided by N, where R is the remainder of P divided by N. 
It is evident that the process will terminate whenever We ob- 
tain r,=0; but when this is not the case, the quotients and re- 
